Download Disk Galaxies and problem 3

–1–
3.
Disk Galaxies
In this section, we shall build on our knowledge of the Milky Way to study the properties of
disk galaxies as a population. Disk galaxies have many components, including
• a disk which contains metal rich stars, gas (HI, molecular hydrogen, dust and hot gas), with
strong ordered rotations.
• a bulge which contains metal-poor to metal-rich stars that show diverse properties. They
show weaker signals of rotations and are strongly affected by the bars if they are present.
• various halos made up of metal poor stars, globular clusters, hot X-ray gas etc.
• a dark matter halo which dominates the mass in the outer part.
3.1.
3.1.1.
Photometry of disk galaxies
disks: radial and vertical distributions
The surface brightness as a function of radius can be approximated as
I(R) = I0 exp(−R/Rd ),
(1)
where Rd is the disk scale length and I0 is the central surface brightness. The total luminosity is
given by Ltot = 2πRd2 I0 .
Fig. 1 shows two examples of the surface brightness profile. In the outer part, exponential
profiles fit the data well. The deviations at the central part are due to bulges (see below).
Fig. 1.— Surface brightness profiles of two disk galaxies.
–2–
3.1.2.
bulges
Their light profiles resemble those of elliptical galaxies. They approximately follow the
so-called de Vaucouleurs law or r1/4 law:
I(R) = Ie e−7.67[(R/Re )
1/4 −1]
,
(2)
where Ie is the surface brightness at Re , and Re is the effective radius (also called the half-light
radius) within which half of the total light is contained. The total luminosity is given by
Ltot = 7.22πRe2 Ie .
In practice, some bulges are actually better described as exponetials rather than the r1/4 law.
3.1.3.
Various halos
The volume density of various halos are as follow.
• stellar halo: ρH ∝ r−3.5 , ρH (R ) ≈ ρdisk /100.
• The number density distribution for globular clusters follow ngc ∝ r−3.5 and
ngc (10 kpc) ≈ 10−3 kpc−3 .
• dark matter halo, for more see section 3.2.2.
3.1.4.
gas and dust
The neutral and molecular hydrogen distributions show diverse geometries. Some show
holes in the neutral hydrogen (HI) distribution at their centres. Molecular gas is usually more
concentrated toward the centre.
The neutral hydrogen and molecular hydrogen distributions of the MW are shown in Figure
14 in the overview chapter (§1). In the MW, the HI mass is ∼ 4.3 × 109 M , and nearly 80% lies
beyond the solar circle R0 = 8 kpc. On the other hand, the molecular hydrogen in the MW is
much more compact, nearly 80% lies inside R0 , with a total mass of about 109 M . Notice that for
the MW, the gas mass is much smaller than the total stellar mass (M∗ ≈ 7 × 1010 M ), i.e., most
of the gas has been converted into stars.
Dust lanes in many spiral galaxies are very striking. However, the effect of dust becomes less
important as the wavelength becomes longer; in the radio, its effect is negligible.
3.2.
Kinematics of disk galaxies
3.2.1.
rotation curves
Most gas follow ordered circular motions around galactic centres. The rotation curve refers
to the run of circular velocity as a function of radius. The atomic hydrogen is often much more
–3–
Fig. 2.— optical image and neutral hydrogen (HI) map (contours) for NGC 3198. Notice that the
HI distribution is much more extended.
extended than the optical light distribution (see Fig. 2), and so they can be used to determine the
rotation curves of disk galaxies to much larger radii.
Fig. 3 shows the rotation curves for many spiral galaxies. There are a few interesting trends
in the shapes of rotation curves:
• Bright (high circular velocity) spirals tend to show slowly declining rotation curves.
• Faint (low circular velocity) spirals tend to show (more) slowly rising rotation curves.
• Rotation curves for Sa type galaxies rise more quickly in the inner part, which is likely due
to their more massive bulges at the centre.
Fig. 3.— Rotation curves for many spiral galaxies.
–4–
3.2.2.
Dark Matter profiles for flat rotation curves
Consider a test particle in circular motion:
mV 2
GMr m
=
, V = const.,
r2
r
(3)
where Mr is the mass enclosed with the radius spherical r.
Eq. (3) implies that Mr rises linearly as a function of radius, i.e., most mass is in the outer
part. But the surface brightness drops exponentially with radius, so most light is in the inner
part. In other words, dark matter dominates in the outer part.
We can actually infer the dark matter density profile. Consider a spherical shell of radius
from r to r + dr, then
dM = ρ(r) dV = ρ(r) 4πr2 dr → ρ(r) =
1 dM
1 V2
=
.
4πr2 dr
4πG r2
(4)
We can integral along the vertical direction to obtain the surface density. We write the cylindrical
coordinates as (R, θ, z), and
Z ∞
V2
dz ρ(R, z) =
Σ(R) =
(5)
.
4GR
−∞
The surface brightness profile scales as I(R) = I0 exp(−R/rd ), and so the mass-to-light ratio
increases as a function of radius, implying a large amount of dark matter in the outer part.
Rotation curves are only approximately flat. Numerical simulations finds the so-called
Navarro, Frenk and White profile
1
ρ(r) ∝
,
(6)
r(r + rs )2
where rs is a scale radius (≈ 20 kpc for the MW), which has to be found from computer simulations.
The profile scales as r−1 at small radius, but as r−3 at large radius. There is an intermediate
range (r ∼ rs ) where the density follows r−2 .
There may be an intriguing disk-halo conspiracy. Fig. 4 shows the rotation curve for
NGC3198. It is puzzling that the stellar mass (more precisely, gas+stars) plus the dark matter are
coordinated such that the rotation curve is approximately flat. Why is this the case (conspiracy)?
The central difficulty is that although we know how much light is emitted in a disk galaxy, we
often do not know its (associated) mass well. Hence we do not know the relative contributions due
to the stars and dark matter in the central part of spiral galaxies.
3.2.3.
Tully-Fisher Relation
The Tully-Fisher relation is a tight correlation between the (maximum) circular velocity and
the total luminosity, with a small scatter. Fig. 5 shows the observed Tully-Fisher relation in four
bands.
β , where L is the total luminosity and
The Tully-Fisher relation can be written as L ∝ Vmax
Vmax is the maximum circular velocity:
–5–
Fig. 4.— Rotation curve for NGC 3198. One possibility of the relative contributions due to the
dark matter and disk is indicated.
• β ≈ 3 in B-band (λ ≈ 4000Å) β ≈ 3.2 in I-band (λ ≈ 8000 Å) β ≈ 4.2 in H-band
(λ ≈ 12, 000Å)
• In the B-band, the total light is sensitive to the presence of young stars, and hence the
Tully-Fisher relation in this band is affected more by dust extinction. While, in the infrared,
the luminosity measures the contribution from all stars, and is also less affected by the dust
extinction. So the infrared is probably a better band to apply the Tully-Fisher relation.
• The Tully-Fisher relation is puzzling because it shows that the total light is tightly correlated
with the circular velocity. But as we haven seen, individual star formation sites are very
complex. Nevertheless, it seems that the total light in a disk galaxy is tightly self-regulated.
The Tully-Fisher relation is an important extra-galactic distance indicator. We can observe
the circular velocity (through redshift of emission lines) and hence infer the total luminosity, L.
Combined with the observed flux (f ), we can infer the distance by d = [L/(4πf )]1/2 .
3.3.
Structures in disk galaxies
3.3.1.
Spiral arms
Spiral arms are the most visually striking features in disk galaxies. They divide spirals into
two classes:
• Grand-design spirals have long and two symmetric arms (∼ 10% of spirals).
• Flocculent spirals have short and fragmented spiral arms, with no apparent symmetry (∼
90% of spirals).
Spiral arms are bluer and younger than the overall disk; they are sites of active star formation.
We return to the origin of spirals in section 3.5.
–6–
Fig. 5.— The Tully-Fisher relation in four different bands. The quantity WRi plotted on the
horizontal axis is roughly equal to 2Vmax .
3.3.2.
Bars/bulges
Bars are very common, occurring in more than 30% of galaxies. See Fig. 8 in the overview
section. They often show straight and rectangular shape and are thought to be rotating.
Early-type spiral galaxies have the strongest bars; they often have constant surface brightness
across the bar. In late-type spirals (flocculent spirals), the bars are weaker, and the surface
brightness profile is more exponential.
3.3.3.
Rings
See Fig. 11 in the overview section for examples of rings in galaxies.
3.3.4.
Warps
Fig. 13 in the overview section shows two examples of warped galaxies. Warps often start at
the end of “visible” disks. For explanations of warps, see the book by Binney & Tremaine (1987).
–7–
3.4.
Dynamics in disk galaxies
In order to fully understand the structures in disk galaxies, we must understand how stars
and gas move in the disk gravitational potential.
Let us study the vertical, radial and azimuthal motions in turn. As disk galaxies are
axis-symmetric systems, it is convenient to use a cylindrical coordinate system (r, θ, z). In this
coordinate system,
~ = r~er + z~ez
• position vector: R
~ = ṙ~er + rθ̇~eθ + z~ez
• velocity: V
• acceleration: ~a = (r̈ − rθ̇2 )~er + (2ṙθ̇ + rθ̈)~eθ + z̈~ez
The mean gravitational potential is, due to symmetry, independent of the azimuthal angle θ,
and can be written as Φ(r, z). The gravitational force (for a unit mass) is given by F~ = −∇Φ(r, z).
3.4.1.
Vertical motions
We consider the motion of a star slightly perturbed close to the mid-plane. As can be easily
shown, the force is always in the opposite direction of the displacement, so the star follows a
simple harmonic motion if the displacement is small.
Some algebra yields
z̈ = −4πGρ0 z ≡ −ωz2 z, ωz ≈ (4πGρ0 )1/2 ,
(7)
where ρ0 is the density in the mid-plane. For the solar neighbourhood, ρ0 ≈ 0.1M /pc2 , the
vertical oscillation period is given by Tz = 2π/ωz = 8.4 × 107 years. This should be compared
with the period for a complete circular orbit in the solar neighbour around the Galactic center of
T = 2.2 × 108 years.
3.4.2.
Radial and azimuthal motions
Stars in a disk galaxy are on nearly circular orbits. Their motions can be approximated as a
circular motion around a guiding center plus an epicycle motion around the guiding center.
Suppose a particle is initially on a circular orbit at radius r0 and has angular velocity ω0 . In
the rotating frame with angular velocity ω0 , the particle is stationary, with the gravitational force
balanced out by the centrifugal force.
If we displace the particle along the azimuthal direction, then the polar angle of the particle
changes a bit but it still stays on the same circular orbit. The particle is called neutrally stable,
and the oscillation frequency is zero.
Let us now consider the motion of the particle if we give the particle a slight radial
perturbation. Notice that angular momentum is conserved, i.e., Lz = ω0 r02 = ωr2 is a constant. If
–8–
the perturbed radius is smaller than r0 , the angular velocity will increase, so in the rotating frame,
the particle will lead the guiding centre. As the radius decreases, the gravitational force increases
as GM/r2 . The centrifugal force (= ω 2 r) increases as L2z /r3 , at a rate faster than the gravitational
force. Hence the particle is forced to move out. The situation is exactly reversed when r > r0 .
From this consideration, it is clear that the particle makes a simple harmonic motion in the
radial and azimuthal directions. The motion can be described by an epicycle with an opposite
sense of rotation compared with the circular orbit.
The frequency of the oscillation is called epicycle frequency, and is given by
∂ω 2 (r) κ2 = 4ω02 + r0
,
∂r r=r
(8)
0
where ω0 and ω(r) are the angular velocities at radius r0 and r, respectively. Clearly the epicycle
frequency depends on the rotation curve shape through ω(r). The axial ratio of the tangential
maximum displacement to the maximum radial displacement is given by 2ω0 /κ.
Using eq. (8), it is easy to show that
• for a rigid body rotation, ω(r) = const., κ = 2ω0
• for a Keplerian rotation, ω(r) ∝ r−3/2 , κ = ω0
√
• for a flat rotation curve, ω(r) ∝ r−1 , κ = 2ω0
• for the Milky Way, observationally we find that κ/ω0 = 1.3 ± 0.2, close to the value expected
for a flat rotation curve.
Fig. 6 shows examples of epicycle motions for a Keplerian potential, solid-body rotation and
for the Milky Way. For the Milky Way, the orbit is not closed as κ/ω0 = 1.3. It takes about
1.7 × 108 yr to finish one radial oscillation the epicycle has an axis ratio of about 1.5.
Fig. 6.— Epicycle patterns. left: for Keplerian potential, κ = ω0 . Middle: for solid-body rotation,
κ = 2ω0 . Right: epicycle for a star at the solar radius, κ ≈ 1.3ω0 . The thin lines indicate the
original circular orbits.
–9–
3.4.3.
Resonances
Stellar orbits, like drums and bridges, have natural frequencies (e.g., the epicycle frequency κ)
and hence can have resonances. In such cases, small perturbations can evolve to large-amplitudes.
The driving force in a disk galaxy can be spiral arms and bars; these perturbations often
rotate with some angular pattern speed, Ωp .
Let us consider a potential perturbation φ1 (R, θ, t), where R is the cylindrical radial
coordinate. The simplest and most important perturbations (like bars) are stationary in a rotating
frame, so the perturbation can be written as φ1 (R, θ − Ωp t). We can expand it in Fourier modes
φ1 (R, θ − Ωp t) =
X
Am cos [m(θ − Ωp t)] .
(9)
The driving frequency is given by d[m(θ − Ωp t)]/dt = m(ω − Ωp ), where ω = dθ/dt is the angular
velocity. Resonances occur when the driving frequency is equal to the intrinsic frequency (0 and κ)
• m(ω − Ωp ) = 0. Star co-rotates with the perturbation, and hence experiences persistent
perturbation
• m(ω − Ωp ) = κ, star experiences successive crests of the perturbation at its natural radial
oscillation frequency. Inner Lindblad resonance.
• m(ω − Ωp ) = −κ. Outer Lindblad resonance.
It can be shown that rings can form at the resonance radii and bars often end at the co-rotation
radius. Furthermore, one theory postulates that peanut-shaped galaxies may be associated with
vertical resonances. So resonances play fundamental roles in many features in spiral galaxies.
3.5.
Origins of Spiral Arms
As we have already discussed, spiral galaxies can be divided into grand-design spirals and
flocculent spirals, with the latter being much more common.
From Sa to Sc galaxies, the spiral arms become more open. The pitch angle becomes larger,
increasing from roughly 10◦ to 18◦ .
Another remarkable fact is that most spiral arms are trailing. This can be easily understood
by the differential rotation and shearing in the disk (see below).
3.5.1.
Winding dilemma
If spiral arms are permanent material arms, due to differential rotation, the spiral arms will be
too tightly wound up compared with observations. This problem is called the “winding dilemma.”
This can be quantitatively understood as follows. Let us consider a straight arm initially on
the x-axis. The angle will be θ(r) = ω(r)t, and the pitch angle is
cot i = |
rdθ
dω
| = |rt |
dr
dr
(10)
– 10 –
For a flat rotation curve, ω = v/r, and hence | cot i| = vt/r. For v = 220km/s, r = 8 kpc, and
t = 1010 yr, we find i = 0.2◦ . This is roughly two orders of magnitude smaller than the observed
values of about 10◦ for Sa galaxies and 18◦ for Sc galaxies.
So spiral arms are not permanent features due to the winding dilemma. They are density
waves: they are made up of different stars at different time! This is like a traffic jam when a road
block is set up, the car flow pattern is a constant as a function of time, but the traffic jam is made
up of different cars at different time.
Fig. 7.— From simulations by Dr. J. Barnes. Two-armed spirals are easily generated by tidal
interactions. Such striking grand-design spirals are observed in M51 and M81.
Fig. 8.— Simulations of chaotic spiral arms, from Gerola & Seiden (1978). The time is in units
of 15 million years. The top and bottom panels are appropriate for two different rotation curves
corresponding to M101 and M81, respectively.
– 11 –
3.5.2.
Theories of spiral arms
There are currently several theories of spiral arms, including
• Tidal origin of spiral arms. Spiral arms are induced by tidal perturbation of a nearby galaxy.
This is observationally seen in grand-design spirals, such as M51. The tidal arms are also
easily produced in numerical simulations, see Fig. 7.
• Bar spiral arms. A rotating bar at the centre of galaxies can also induce spiral arms that
can last for a long time. This may be another way to form grand-design spirals.
• Chaotic spirals arms theory. A simulation for this theory is shown in Fig. 8. In this theory,
stars form in the disk due to local gravitational instabilities. Differential rotation shears the
star formation sites into spiral patterns. As time passes, the arms are sheared more and the
brightest stars die. Both effects lead to the disappearance of the gradual disappearance of
the arm fragment. But star formation occurs elsewhere. This theory may be appropriate for
flocculent spirals, Such spirals have many short, fragmented arms, with no clear two-arm
spiral symmetry.
In all these theories, swing amplification plays an important role, converting any leading arms into
trailing arms, and amplifying the spiral amplitudes in the process (see Toomre 1981).
Problem Set 3
1. The mass density profile of a (spherical) galaxy is given by
ρ(r) =
V2
,
4πGr2
where r is the spherical radius. Show that
a) the rotation curve for this galaxy is flat as a function of radius.
b) its surface density is given by
V2
,
Σ(R) =
4GR
where the cylindrical coordinates are denoted by (R, φ, z).
2. An edge-on spiral galaxy has a maximum circular velocity (Vmax ) of 200 km s−1 , and its
apparent I-band magnitude is 18.5. The I-band Tully-Fisher relation is given by
MI − 5 log h ≈ −21.00 − 7.68 [log(2Vmax ) − 2.5],
where h = 0.7 and MI is the absolute magnitude. Using the above Tully-Fisher relation to
estimate the distance to the spiral galaxy.
3. Can we use face-on disk galaxies to study the Tully-Fisher relation?
4. The epicycle frequency (κ) at a radius r0 is given by
2
κ =
4ω02
∂ω 2 (r) ,
+ r0
∂r r=r
0
where ω0 and ω(r) are the angular velocities at r0 and r, respectively. Show that for a flat
√
rotation curve κ = 2ω0 .
5. The rotation curve of a galaxy can be approximated as
V (r) = 200 km s−1
=
r
1 kpc
200 km s−1 ,
, r < 1 kpc
r > 1 kpc
(11)
The galaxy has a bar at its centre, with a pattern speed of Ωp = 40 km s−1 kpc−1 . Using the
epicycle frequency given in the previous problem,
a) write down the conditions for the resonances
b) find the co-rotation radius and the radii for the inner and outer Lindblad resonances.