Download –1– 2. Milky Way We know a great deal, perhaps more than any

–1–
2.
Milky Way
We know a great deal, perhaps more than any other galaxy, about the Milky Way (MW)
due to our proximity. However, our inside position also hampers our understanding of the Milky
Way. Fore example, the spiral arms of the Milky Way have to be inferred, while for many external
galaxies, we can often directly see these features. Another difficulty is that, at least in the optical,
the Milky Way is heavily obscured by dust in the Galactic plane.
Many questions about the Milky Way are still not completely understood. To name a few, we
still do not completely understand the formation of the MW and the physical parameters of the
Galactic bulge. We will discuss the formation of disk galaxies such as the Milky Way in Chapter
§5 on Galaxy Formation.
2.1.
Structure of the Milky Way: a brief history
On a clear night, the Milky Way appears to like a band with dark patches. Greeks and
Romans thought the Milky Way was like a fluid (Greeks: Galaxies Kuklos, and Romans: Via
Lactea). In 1610, Galileo resolved the Milky Way into numerous stars.
The structure of the MW was not properly understood until 1920’s. Our position inside the
Milky Way made it more difficult to infer the spatial structure of the Milky Way. We only see
the angular distribution of stars on the sky, as we do not know the distance to stars (particularly
distant stars), so it is difficult to convert the two-dimensional angular distribution into a spatial
distribution.
Fig. 1.— Panorama of the Milky Way (credit: Lund Observatory). Notice the dark patches caused
by dust extinction.
A Panorama of the Milky Way is shown in Fig. 1. The Galaxy appears as a thin band with
dark patches which are caused by dust extinction. Many past astronomers often ignored such
obvious dust extinctions! Under this assumption, if one further assumes that all stars have similar
intrinsic luminosities, then their distances would be over-estimated, and the wrong structure for
the MW would be inferred.
–2–
2.1.1.
Photometric Model
• Herschel (1785) used star counts to infer a flattened distribution for the MW.
• A similar, but much larger survey of nearby stars was done by Kapteyn around 1920. He
used parallax, proper motions, radial velocities and spectra to infer the distance to stars. He
inferred that the size of the MW is about 10 kpc, and the MW is flattened with an axial
ratio of 1/5. The Sun is about 650 pc offset from the centre.
Kapteyn was aware of extinction problems. He searched for these, but did not find any
significant effect. However, the mistake he made was at that time he was only aware of
scattering by dust particles, but he did not consider absorption, which turns out to have a
much more important effect.
• Shapley in 1920’s studied globular clusters and used the variables inside them as distance
indicators. He found that globular clusters show a spherical distribution centred on a point
15 kpc away from the Sun (Fig. 2). He argued that globular clusters are massive systems
made up of 105 − 106 stars and hence they should trace out the structure of the MW. He
reached the conclusion that the center of the MW is not at the solar position.
Shapley ignored the effect of dust. So in order to explain Kapteyn’s star counts, he had to
add a local over-density of stars.
Fig. 2.— Distribution of globular clusters in the MW.
• The effect of dust was understood by Trumpler (1930) who used the angular size of open
clusters as a distance indicator. If one assumes that open clusters have roughly the same
luminosity, then one can estimate how bright the cluster would be without extinction. It
turns out open clusters are much fainter than such inferred values. The difference is then
attributed to the dimming caused by dust.
–3–
2.1.2.
Kinematic model of the MW
• Kant (1755) considered the MW as a rotationally supported system.
• Lindblad (1927) realised stars with high velocities (v ∼ 250 km s−1 ) can not be bound by the
Kapteyn’s universe as its mass is too small. On the other hand, these stars cannot escape
from the Shapley’s universe as it has a much higher mass.
• Oort (∼ 1927) completed the picture. In this picture the high velocity stars and globular
clusters form a different population (so called halo population, or pop II). They do not follow
the Galactic rotation, and their spherical distributions are caused by their large random
motions. In contrast, disk stars and open clusters follow the Galactic rotation and belong to
the disk population (or pop I).
• The atomic hydrogen (HI) 21 cm emission line was predicted by Van de Hulst in 1944 and
observed by Ewan and Purcell in 1951. It was soon realised that this line can used to map
the MW. These observations indicate non-circular motions of the gas which were found
to be due to perturbations of spiral arms. The central HI gas, in particular, shows large
non-circular motions due to the Galactic bar (realised in 1970’s).
Fig. 3.— A schematic face-on view of the MW.
Fig. 3 shows a schematic picture of the MW if it is viewed face-on. MW is now classified as
an SBc or SBb spiral galaxy.
–4–
2.2.
Photometric model of the MW
2.2.1.
Galactic disk
The surface brightness (luminosity per unit area) is approximately an exponential:
(1)
I(R) = I0 exp(−R/Rd ),
where I0 is the central surface brightness and Rd is called the scalelength.
The total luminosity is:
Z ∞
Ltot =
0
I(R) 2π R dR = 2πI0 Rd2
(2)
For the MW, Kent et al. (1991) found in the K-band that Rd ≈ 3 kpc, ΣB ≈ 1000LB, / pc2 , and
Ltot ≈ 4.9 × 1010 L .
Using star counts from the SLOAN digital sky survey, Chen et al. (2001) found that the Sun
is about 27 pc above the Galactic plane. Furthermore, the vertical distribution of stars in the
solar neighborhood can be approximated by the sum of two exponentials. These were identified
as a thin disk and a thick disk components. The stars in the thin and thick disks appear to have
different ages and metallicities (see the Table below).
component
thin disk
thick disk
scale height (pc)
z0 = 330
z0 = 580 − 750
mass (M )
5 × 1010
∼ 109
2.2.2.
Vc ( km s−1 )
220
180
Metallicity (Z )
0.5-2
0.3-0.5
age (Gyr)
<10
>10
∼
Galactic bulge (bar)
There is now substantial evidence for a Galactic bar
• HI observations indicate larger non-circular motions around the Galactic centre.
• Infra-light observations indicate asymmetry with longitude (see Fig. 4).
• Stars with near constant luminosity (standard candles) appear brighter on the positive
longitude side (see Fig. 3).
Unfortunately, the density distribution of the Galactic bulge is still rather uncertain.
• The total luminosity is about 1010 L in the K-band, uncertain by a factor of ∼ 2.
• The axial ratios are about 1.7 kpc : 0.6 kpc : 0.4 kpc. The minor axis coincides with the
vertical axis, but the major axis is about ∼ 15◦ away from line of sight. The near-side of the
bar is at positive longitude (the first quadrant).
• The size of bar is a few kpc, and the bar rotates with a pattern (angular) speed of
Ωp = 60/ Gyr with a period of ∼ 108 yr. The bar causes non-circular motions and resonances
in the stars and gas out to ∼ 4 kpc.
–5–
Fig. 4.— Contours of the infrared light emission detected by the COBE satellite. Notice the
asymmetry with longitude. This is due to the near-side of the bar being located in the first
quadrant (positive longitude).
2.3.
Kinematics of MW
Motions of stars and gas are important to understand the structure and mass distribution
(including dark matter) in the MW. They also give strong hints about how the MW forms and
evolves.
Most stars and gas follow the Galactic rotation together with some random motions. In
practice, we can measure the radial velocity (Vr ) of stars using Doppler shifts and the proper
motions µ (usually measured in arcsec/yr). The proper motion is equal to Vt /d, where Vt is the
transverse velocity and d is the distance. If we know the distance, then we can measure Vt (which
is a vector and has two components), and hence we can measure all the three components of the
velocity.
The Galactic coordinate system is centred at the Sun, and the (x, y, z) is usually chosen such
that the x-direction is toward the Galactic centre, the y-direction toward the Galactic rotation,
and the z-direction is toward the North Galactic Pole, chosen to form a right-handed system. The
angular coordinates of stars are labelled as (l, b), and the velocities are also often indicated by
(U, V, W ). The line of sight toward the Galactic centre has l = 0.
2.3.1.
Oort constants: rotation in the solar neighborhood
The differential rotation of the MW implies that stars exhibit both radial, tangential and
vertical motions. By studying these motions, we can infer the rotation curve of the Galaxy.
We will assume that all stars rigourously follow the circular rotation of the Galaxy. The radial
and tangential motions of an object in the Galactic plane at radius R from the Galactic centre are
Vr = R0 sin l [ω(R) − ω0 ] , Vt = R0 cos l [ω(R) − ω0 ] − ω(R)d,
(3)
where ω(R) and ω0 a1re the angular velocities at radius R and R0 , l is Galactic longitude, and d
is the distance from the Sun to the object. Notice that these formulae are valid for any R.
–6–
In the solar neighbourhood, we can use approximations that are correct to first order of d,
R ≈ R0 − d cos l,
and
(4)
dω ω(R) − ω0 =
(R − R0 ),
dR R0
One can show that
1
dω =
dR R0
R0
dV V0
−
dR R0 R0
!
≡−
(5)
2A
,
R0
(6)
where A is Oort constant A. Using these relations, we find that
Vr = A d sin(2l),
Vt = d [A cos(2l) + B],
(7)
where the Oort constants A and B are defined by
1
A=
2
V0
dV −
R0
dR R0
!
and
1
B=−
2
dV V0
+
R0
dR R0
(8)
!
(9)
Key points concerning the Oort constants:
• They can be determined by local measurements
• For flat rotation curves, i.e., V (R) =const., A = ω0 /2 = −B and hence A + B = 0. For
solid-body rotations, i.e., ω(R) = ω0 =const., A = 0, B = −ω0 . Notice that there is no radial
velocity for a solid-body rotation.
• Radial velocities can be measured using Doppler shifts, while tangential velocities can be
measured using proper motions
µl =
δθ
Vt
=
= A cos(2l) + B,
δt
d
(10)
which is independent of d. The proper motions are usually measured in arcseconds per year.
• The Hipparcos satellite has measured proper motions accurately for many
nearby stars. Using these proper motions, Feast and Whitelock (1997) found:
A = (14.8 ± 0.8) km s−1 kpc−1 , B = (−12.4 ± 0.6) km s−1 kpc−1
• These constants imply that the rotation curve is about 218 km s−1 for R0 = 8 kpc (see below)
and slowly falling (see Problem set 2), in fact V ∝ R−0.088 .
Distance to the Galactic centre
The distances to the Galactic centre can be determined using a number of tracer populations.
For example, the centre can be determined using the density peak of globular clusters, see Fig. 2.
Observationally, we find from globular clusters that
R0 = (8.0 ± 0.8) kpc,
the distance determined using this method has a 10% uncertainty.
(11)
–7–
2.3.2.
Rotation in the inner galaxy (R < R0 )
The rotation curve of the Milky Way can be probed using both stars and gas. HI (atomic
hydrogen) is particularly important due to its 21cm emission.
The ground state of the H-atom is split into two hyperfine levels by the coupling between the
spins of the electron and proton; the atom’s energy is 6 × 10−6 eV higher when the two spins are
parallel than that when they are anti-parallel.
The transition between these two hyperfine levels is “forbidden”, and has a wavelength of
21cm. As hydrogen is the most abundant element in the universe, HI emission can be observed in
many places, including the MW. We can map the HI intensity as a function of longitude, latitude
and radial velocity.
In a given direction, the HI intensity is a function of the radial velocity. For lines of sight
with 0 < l < 90◦ , the intensity drops rapidly beyond a certain maximum radial velocity. This is
because the angular velocity usually decreases as a function of radius, which immediately implies
that the radial velocity is the largest at the tangent point when R = R0 sin l. And the radial
velocity at the tangent point is given by
Vc (R0 sin l) = Vr(t) + V0 sin l,
(12)
(t)
where Vr is the radial velocity at the tangent point. Using this method, we can in principle
determine the rotation speed for radius between 0 < R < R0 .
2.3.3.
Rotation curve in the outer Galaxy (R > R0 )
The Outer rotation curve cannot be determined using the method discussed above as there
is no tangent point for 90◦ < l < 270◦ . However, we can use a population of tracers with known
distances. The distance between the tracers and the Galactic centre can be determined from
R2 = R02 + d2 − 2R0 d cos l.
(13)
We can also determine the radial velocity of the tracers, then ω(R) and hence V (R) can be
inferred using eq. (3). A prominent tracer population is Cepheid variables. They follow a
so-called luminosity-period relation. These stars are very bright and so they can observed to large
distances. Their periods can be measured from their periodic variable light curves. For a Cepheid
variable, we can first determine its period, and then infer its intrinsic luminosity. Combined with
the observed flux, we can infer its distance. Other tracer populations include globular clusters,
planetary nebulae and Carbon stars. Fig. 5 shows the rotation curve out to roughly 20 kpc.
Using this method, the Galactic rotation curve can be determined out to R ∼ 20 kpc.
We can use halo stars and satellite galaxies to determine the mass distribution at even
larger distances. From the motions of these satellite galaxies (see Fig. 6), we can infer the mass
distribution as a function of radius, which can then be used to infer the rotation curve. Using this
method, the rotation curve of the Milky Way is found to be flat all the way out to about 200 kpc.
–8–
Fig. 5.— The rotation curve for the Milky Way.
150 km s−1 , not from zero.
Notice that the vertical scale starts from
Fig. 6.— A dozen or so Satellite galaxies in the Milky Way. The bar at the top of the figure shows
the scale for 105 light years.
–9–
2.4.
Dark matter in the MW
Dark matter distribution in the MW is still not completely understood. We shall consider the
dark matter distribution in the solar neighbourhood, in the inner galaxy (for R < R0 ) and in the
outer galaxy (for R > R0 ).
2.4.1.
Dark matter in the solar neighbourhood
Kinematics of nearby stars yield
Σtot (|z| < 1.1 kpc) = (71 ± 6)M / pc2 .
(14)
Σ∗ = 26.9M / pc2 .
(15)
Σgas = 13.7M / pc2 .
(16)
HST star counts give
And the gas surface density is roughly
And hence the dark matter surface density is given by
ΣDM = Σtot − Σ∗ − Σgas = (30 ± 15)M / pc2 .
(17)
Notice that the error bar is quite large. Around the solar neighbourhood, there is weak evidence
for dark matter.
2.4.2.
Dark Matter in the inner galaxy (R < R0 )
Recall that in the K-band (see §2.2.1) the Galactic disk has a central brightness of
I0 ≈ 103 L pc−2 and a disk scalelength of Rd = 3 kpc, so for the solar neighbourhood, the
mass-to-light ratio in the K-band is given by
M
Σtot
71M pc−2
M
=
=
≈1
.
L
I
I0 exp(−R0 /Rd )
L
(18)
If we assume the mass-to-light ratio is the same throughout the Galactic disk, then the total disk
10
mass is about Mdisk = Ltot M
L = 4.9 × 10 M . Similarly, the total mass of the bulge is about
Mbulge = 1.5 × 1010 M ,
(19)
which is uncertain by ∼ 40%.
For a spherical mass distribution, we have for the solar neighbourhood1 ,
Vc2 (R0 ) =
1
G(MDM + Mdisk + Mbulge )
V 2 R0
→ MDM = c
− Mdisk − Mbulge ,
R0
G
(20)
We ignored the complication that for a flattened disk system, its circular velocity diffs from that of a spherical
distribution.
– 10 –
where MDM , Mdisk and Mbulge are the dark matter, disk and bulge masses within R0 , respectively.
And hence
V 2 R0
MDM = c
− Mdisk − Mbulge ≈ 2.6 × 1010 M .
(21)
G
So within the solar circle, the fraction of dark matter is about 30%, although this number is still
quite uncertain. Most of the mass (≈ 70%) within the solar circle is therefore made up stars and
gas. Of course, baryons (protons and neutrons) dominate the mass of stars and gas (the mass of
electrons is negligible), so we can say that most of the mass within the solar circle is in baryons.
2.4.3.
Dark Matter in the outer galaxy (R > R0 )
The mass distribution can be inferred using halo stars, tidal streams and satellite galaxies.
These studies indicate that the total mass is roughly a linear function of radius
M = 2 × 1012 M
r
.
200 kpc
(22)
One interesting question is that within 200 kpc, the baryon fraction is only of the order
fbaryon =
Mdisk + Mbulge
≈ 3.2%,
M
(23)
much lower than the average baryon fraction (16%) in the universe from, e.g., observations of
the microwave background observations. It appears that the Milky Way is quite inefficient in
gathering baryons.
Fig. 7.— Sagittarius dwarf galaxy, angular (left panel) and spatial positions (right panel). This
dwarf elliptical galaxy is is being ripped apart by tidal forces into long streams of stars that will
eventually be merged into the Milky Way.
One important discovery in the last decade or so was the Sagittarius dwarf galaxy. This galaxy
was initially discovered just behind the centre of the MW (see Fig. 7). Infrared-observations of
M -giants further reveal a stream of stars associated with this galaxy (see Fig. 8). The dwarf
galaxy is being tidally disrupted by the intense gravitational field of the MW and stars are being
stripped off the galaxy which produce the tidal streams. The stars are observed to be in a plane
almost pole-on with a tilt of about 15◦ .
– 11 –
Fig. 8.— Sagittarius dwarf galaxy tidal streams. Another view, from Martnez-Delgado et al (2003).
Computer simulations can reproduce the observed stream (see Fig. 9). The stars in the
Sagittarius stream seem to be distributed in a plane. If the halo is highly flattened, then the stars
will precess due to torques and hence can not stay in the same plane for a long time. The fact
that they are more or less in the same plane implies that the dark matter distribution (which
dominates the mass budget) is quasi-spherical.
Fig. 10 shows a MW-type halo found in numerical simulations. There are many striking
substructures in the simulations. The number of substructures seems to exceed the number (∼
a dozen) of luminous satellite galaxies we observe in the MW. It is unclear at present how this
discrepancy (if there is one) can be resolved.
Summary: components of the Milky Way
The following table summarises all the components of the Milky Way.
component
thin disk
thick disk
bulge
globular clusters
stellar halo
dark matter halo
HI
H2
size (kpc)
rd ∼ 2 − 3
z0 ≈ 0.3
z0 ≈ 0.6
∼ few
∼tens
< 200
< 200
< 20
∼5
mass (M )
5 × 1010
109
∼
(1 − 2) × 1010
few ×107
< 109
2 × 1012
4.3 × 109
∼ 109
Vc ( km s−1 )
220
180
∼ 100
∼ 40
∼ 40
∼ 40
220
220
Metallicity (Z )
∼ 0.5-2
∼ 0.3-0.5
∼ 0.5-2
∼ 0.1
∼ 0.1
–
–
–
age (Gyr)
<
∼ 10
>10
∼
mostly old
>10
∼
>10
∼
–
–
–
Notice that the metallicity (commonly written as Z) refers to the fractions of mass in elements
heavier than hydrogen and helium. The solar metallicity (Z ) is approximately 0.02 and this is
often used as the units.
– 12 –
Fig. 9.— Sagittarius dwarf galaxy: computer simulations by Dr. K. Johnston. The satellite moves
counter-clock-wise in this figure.
The ages and metallicities of different populations in the MW provide clues how different
components were assembled. We will return to this issue when we discuss the theoretical models
of galaxy formation.
– 13 –
Fig. 10.— A Milky Way type dark matter halo in computer simulations. Notice the large number
of substructures present.
Galaxies: Problem Set 2
1. Revision questions: how do we know that a) the Milky Way has spiral arms? b) the Sun is
about 8kpc from the centre of the Milky Way? c) the Milky Way has a bar at the centre?
d) the Milky Way has a massive dark matter halo? e) Milky Way has dust extinction?
2. What is the typical angular size of a Milky-Way sized galaxy at a distance of 1Gpc? Can
the Hubble Space Telescope (with a resolution of about 0.05 arcsecond) resolve it?
3. What is the acceleration experienced by the Sun in its orbit around the Galactic centre?
How does that compare with acceleration experienced by the Earth around the Sun?
4. Given the definitions of Oort’s constants A and B
1
A=
2
V0
dV −
R0
dR R0
!
and
1
B=−
2
V0
dV +
R0
dR R0
!
where R0 (= 8kpc) and V0 are the distance to the Galactic centre and the rotation velocity,
show that the rotation curve falls off as V ∝ rβ , where β = −(A + B)/(A − B). What is the
β-value for the solar neighbourhood?
5. Consider the density profile (so-called Navarro, Frenk and White profile) of a dark matter
halo found in computer simulations with
ρ=
A
,
r(r + rs )2
for r < 200kpc
– 14 –
where the scale radius rs is 20 kpc.
• Integrate and find its mass within radius r
• Assuming the rotation velocity is 200 km/s at r = 200 kpc, plot the rotation curve and
comment on its flatness.
6. (From Binney & Merrifield 1998) Show that for any spherical Galactic potential, the escape
velocity from the solar neighbourhood, Ve , and the local circular speed, Vc , are related by
Ve
Vc ≤ √
2
(24)
with equality in the case that there is no mass at R > R0 . Stars are observed that have
Galacto-centric speeds up to ∼ 400 km s−1 . From this observation, obtain an upper limit on
Vc . Why might this upper limit be misleading?
7. (From Binney & Merrifield) The vorticity of a fluid flow ~v (~x) is ∇ × ~v . In the case of a disk
galaxy such as the Milky Way this flow represents the differential rotation of the stars. Show
that locally ω
~ is simply related to the Oort constants by |ω| = 2B.