Astronomy 3500
Galaxies and Cosmology
Examine the nature and morphology of galaxies of
stars and their distribution in space, and to use
that knowledge to understand the nature of the
universe as a whole. Topics include the Milky Way
Galaxy, the various morphological types of galaxies
and their spatial distribution, exotic objects
including active galaxies, and what the
characteristics of galaxies and clusters of galaxies
tell us about the nature of the universe as a whole.
Emphasis is placed on the development of critical
judgment to separate observational information
from proposed physical models.
24. The Milky Way Galaxy
Goals:
1. Summarize the basic observational
characteristics of the Galaxy and how
they were established, distinguishing
observational fact from physical models.
2. Introduce the dynamical equations for
Galactic rotation and what they tell us
about the nature of the Milky Way.
3. Characterize basic features of the Galaxy,
the Galactic centre, the central bulge, the
disk with its spiral features, and the halo.
Picturing the Galaxy:
The proper study of the Milky Way Galaxy probably
begins in 1610, when Galileo first discovered that the
Milky Way consists of “innumerable” faint stars. In 1718
Halley discovered the proper motions of Arcturus, Sirius,
and Aldebaran, and by 1760 Mayer had published
proper motions for some 80 stars based upon
comparisons of their recorded positions. His results
established that the Sun and stars are not at rest relative
to one another in the Galaxy.
The obvious problem with trying to map our Galaxy
from within is that the Sun is but one of many billions of
stars that populate it, and our vantage point in the disk
8-9 kpc from the Galactic centre makes it difficult to
detect objects in regions obscured by interstellar dust.
But attempts have been made frequently.
In 1785 William Herschel derived the first schematic
picture of the Galaxy from optical “star gauging” in 700
separate regions of the sky. He did it by making star
counts to the visual limit of his 20 foot (72-inch
diameter) telescope. He assumed that r ~ N1/3 (i.e. N ~ r3),
and obtained relative thicknesses for the Galactic disk in
the various directions sampled. No absolute dimensions
were established. By 1817 Hershel had adopted a new
picture of the Galaxy as a flattened disk of nearly infinite
extension (similar to the modern picture).
In 1837 Argelander, of the Bonn Observatory and
orginator of the BD catalogue, was able to derive an apex
for the solar motion from studying stellar proper
motions. His result is very similar to that recognized
today. Also in 1837, Frederick Struve found evidence for
interstellar extinction in star count data, which was
considered necessary at that time to resolve Herschel’s
“infinite universe” with Olber’s paradox (which had
been published in 1823).
By the turn of the century many astronomers felt that a
concerted, detailed effort should be made to establish
reliable dimensions for the Milky Way. The task was
initiated by Kapteyn in 1905 with his plan to study in a
systematic fashion 206 special areas, each 1° square,
covering most of the sky — the well known “Selected
Areas” for Galactic research. By then, separatelypursued research programs into the nature of the Milky
Way system often produced distinctly different results.
In 1918, for example, Shapley noted the asymmetric
location of the centre of the globular cluster system with
respect to the Sun, and suggested that it coincided with
the centre of the Galaxy. But the distance to the Galactic
centre found in such fashion was initially overly large
because of distance scale problems.
Longitude distribution
of globular clusters.
Kapteyn and van Rhijn published initial results from
star counts in 1920, namely a Galaxy model with a
radius of ~4.5 kpc along its major plane and a radius of
~0.8 kpc at the poles. Kapteyn published an alternate
model in 1922 in with the Sun displaced from the centre,
yet by less than the distance of ~15 kpc to the centre of
the globular cluster system established by Shapley.
The issue reached a turning point in 1920 with the well
known Shapley-Curtis debate on the extent of the
Galactic system. The merits of the arguments presented
on both sides of this debate have been the subject of
considerable study over the years, but it was years later
before the true extragalactic nature of the spiral nebulae
was recognized. Although Shapley was considered the
“winner” of the debate, it was Curtis who argued the
correct points. A big step was Hubble’s 1924 derivation
of the distance to the Andromeda Nebula using Cepheid
variables. Somewhat less well-known is Lindblad’s 1926
development of a mathematical model for Galactic
rotation. Lindblad’s model was developed further in
1927-28 by Oort, who demonstrated its applicability to
the radial velocity data for stars. Finally, in 1930
Trumpler provided solid evidence for the existence of
interstellar extinction from an extremely detailed study
of the distances and diameters of open star clusters.
Perhaps the best “picture” of the Galaxy is that sketched
by Sergei Gaposhkin from Australia, as published in
Vistas in Astronomy, 3, 289, 1957. The lower view is
Sergei’s attempt to step outwards by 1 kpc from the Sun.
Sergei Gaposhkin’s drawing is crucial for the insights it
provides into the size and nature of the Galactic bulge,
that spheroidal (or bar-shaped?) distribution of stars
surrounding the Galactic centre. Keep in mind that all
such attempts rely heavily upon the ability of the human
eye (and brain) to distinguish a “grand design” from the
confusing picture posed by the interaction of dark dust
clouds, bright gaseous nebulae, and rich star fields along
the length of the Milky Way (see below).
The present picture of the Galaxy has the Sun lying ~20
pc above the centreline of a flattened disk, ~8.5±0.5 kpc
from the Galactic centre. The spheroidal halo is well
established, but the existence of a sizable central bar and
the nature of the spiral arms are more controversial.
Another schematic representing the present view of the
Galaxy.
An outdated picture of the Galaxy by the instructor
prior to 2010 had the Sun lying ~20 pc above the
centreline of a flattened disk, ~9±1 kpc from the Galactic
centre. The spheroidal halo is well established, and there
is an obvious warping of the Galactic disk in the
direction of the Magellanic Clouds that is best seen in the
fourth Galactic quadrant.
Star Count Analysis:
Define, for a particular area of sky:
N(m) = total number of stars brighter than magnitude m
per square degree of sky, and
A(m) = the total number of stars of apparent magnitude
m ±½ in the same area (usually steps of 1 mag are used).
N(m) increases by the amount A(m)m for each increase
m in magnitude m.

or
dN(m) = A(m) dm,
A(m) = dN(m)/dm.
Star counts in restricted magnitude intervals are usually
made over a restricted area of sky subtending a solid
angle = . The entire sky consists of 4 steradians = 4
(radian)2 = 4 (57.2957795)2 square degrees = 41,252.96
square degrees ≈ 41,253 square degrees. Thus, 1 steradian
= 41,253/4 square degrees = 3283 square degrees.
In order to consider the density of stars per unit distance
interval of space in the same direction, it is necessary to
consider the star counts as functions of distance, i.e. N(r),
A(r). If the space density distribution is D(r) = number of
stars per cubic parsec at the distance r in the line of sight,
then:
r
N ( r )   r 2 D( r )dr . If D(r) = constant = D, then:
0
r
r
N (r )   r Ddr  D  r 2dr  13Dr 3
0
2
0
Cumulative star counts in a particular area of sky should
therefore increase as r3 for the case of a uniform density
of stars as a function of distance. For no absorption:
m – M = 5 log r – 5, 0.2(m – M) + 1 = log r, or r =
10[0.2(m – M) + 1] .
0.6 m M 
Thus, N (m)  13 D 100.2m M 13  1000


D
10
3
0.6 M
0.6 m
0.6 m  C

 1000

D
10
10

10
3
if M and D are constant.
,
i.e.
log N(m) = 0.6m + C.
A(m) = dN(m)/dm = d/dm [10C 100.6m] =
(0.6)(10C)(loge10)100.6m = C'100.6m.
Denote lo = the light received from a star with m = 0.

l(m) = lo10–0.4m [m1–m2 = –2.5 log b1/b2].
or
–0.4 m = log l(m)/lo .
The total light received from stars of magnitude m is
therefore given by:
L(m) = l(m) A(m) (per unit interval of sky)
= loC'10–0.4m + 0.6m = loC'100.2m .
The total light received by all stars brighter than
magnitude m is given by:
Ltot(m) = ∫ L(m')dm' = loC' ∫100.2m'dm' = K 100.2m,
where K is a constant.
Thus, Ltot(m) diverges exponentially as m increases
(Olber’s Paradox).
The results from actual star counts in various Galactic
fields are:
i. Bright stars are nearly uniformly
distributed between the pole and
the plane of the Galaxy, but faint
stars are clearly concentrated
towards the Galactic plane.
ii. Most of the light from the region
of the Galactic poles comes from
stars brighter than m ≈ 10, while
most of the light from the Galactic
plane comes from fainter stars
(maximum at m ≈ 13).
iii. Increments in log A(m) are less
than the value predicted for a
uniform star density, no interstellar
extinction, and all stars of the same
intrinsic brightness.
It implies that D(r) could decrease
with increasing distance (a feature
of the local star cloud that could
very well be true according to the
work of Bok and Herbst), or
interstellar extinction could be
present (or both!). The existence
of a local star density maximum
is also confirmed by the star density
analysis of McCuskey (right).
Actual star counts were done in the past using (m, log )
tables (magnitude, parallax), which were simple to use
with experience. For each value of m, the entries reach a
maximum at some value of log k. The summation of the
entries for each column gives the values for the predicted
counts. The values can be compared with actual star
counts in a particular area, which are usually much
smaller. They must be reduced by the values for the
apparent density function (rk) for each shell. It is
therefore necessary to reconstruct the (m, log ) table
including an estimated (rk) function. A solution for the
observed counts generally requires a number of
iterations with a variable (rk) function until a best
match is obtained. Experience is particularly helpful.
Once a solution for (rk) is obtained, one still needs to
know a(r) to obtain D(r) from the results. Such a(r)
estimates can come from various sources, e.g. Neckel &
Klare (A&AS, 42, 251, 1980).
An Example of a m-log  Table.
As tied to Van Rhijn’s luminosity function.
The use of star counts inside and around the Veil Nebula
in Cygnus (part of the Cygnus Loop) to determine the
distance to the dust cloud and the amount of extinction it
produces at photographic (blue) wavelengths.
Well-recognized characteristics of the Galaxy:
1. Gould’s Belt, consisting of nearby young stars
(spectral types O and B) defining a plane that is inclined
to the Galactic plane by 15 to 20. Its origin is
uncertain. The implication is that the local disk is bent or
warped relative to the overall plane of the Galaxy. This is
not to be confused with the warping of the outer edges of
the Galaxy.
2. An abundance gradient exists in the Galactic disk and
halo, consistent with the most active pollution by heavy
elements occurring in the densest regions of these parts
of the Galaxy. See results below from Andrievsky et al.
A&A, 413, 159, 2004 obtained from stellar atmosphere
analyses of Cepheid variables.
The abundance gradient is also seen in the halo
according to the distribution of globular clusters of
different metallicity relative to the Galactic centre
(below).
3. The orbital speed of the Sun about the Galactic centre
is about 251±9 km s–1, as determined from the measured
velocities of local group galaxies, as well as from a gap in
the local velocity distribution of stars corresponding to
“plunging disk” stars (Turner 2014, CJP, 92, 959). This
fact is actually NOT “well recognized” by most
astronomers.
4. The Galactic bulge is spheroidal, although some
researchers believe it displays a boxy structure at
infrared wavelengths suggestive of a central bar viewed
nearly edge on. A mapping
(right) of Milky Way
planetary nebulae in
Galactic co-ordinates
(Majaess et al. MNRAS,
398, 263, 2009) suggests a
more spheroidal structure
typical of galaxies like
NGC 4565 (top). The
nature of the Galactic
bulge is still unclear. The
surface brightness follows
a de Vaucouleurs law.
 r  4 
 I (r) 
log 10 
  3.3307    1

 re 

 Ie 
1
5. The Galaxy is a spiral galaxy. But does it have 2 arms
or 4, and can it be matched by a logarithmic spiral? A
“grand design” spiral pattern is not obvious in the plot
of the projected distribution of Cepheids (points) and
young open clusters (circled points) below (Majaess et al.
2009).
A schematic representation
of what are considered to
be major spiral features.
How would you connect
the points?
Most recent studies consider
the Cygnus feature to be a
spur or minor arm, and the
Perseus feature is considered
to be a major arm!
There is an “Outer Perseus
Arm” in many deep surveys.
It lies >4 kpc from the Sun
In the direction of the
Galactic anticentre.
6. The Galactic disk is warped, presumably from a
gravitational interaction with the Magellanic Clouds.
The warp is evident in 21cm maps of neutral hydrogen
restricted (by radial velocity) to lie at large distances
from the Galactic centre (below).
7. The Galaxy has a magnetic field that appears to be
coincident with its spiral arms (or features), with the
likely geometry of the magnetic field lines running along
the arms. Weak fields of ~tens of mGauss are typically
measured. The evidence for the presence of a magnetic
field comes from the detection of interstellar polarization
in the direction of distant stars (see below).
8. Note features in the textbook that are NOT included in
the list:
spiral structure
the Milky Way’s central bar
3-kpc expanding arm
dark matter halo
evidence of dark matter
Can you understand why?
Kinematics of the Milky Way:
The Galactic co-ordinate system is defined such that the
Galactic midplane is defined by main plane of 21cm
emission. The zero-point is defined by the direction
towards the Galactic centre (GC), which is assumed to
be coincident with Sagittarius A*.
The Galaxy’s rotation is observed to be clockwise as
viewed from the direction of the north Galactic pole
(NGP). Galactic co-ordinates are Galactic longitude, l,
measured in the direction of increasing right ascension
from the direction of the GC, and Galactic latitude, b,
measured northward (positive) or southward (negative)
from the Galactic plane.
The velocity system for objects in the Galaxy is defined
by:
Θ = Rdθ/dt, the velocity in the direction of Galactic
rotation
Π = dR/dt, the velocity towards the Galactic anticentre
Z = dz/dt, the velocity out of the Galactic plane.
The equations of motion are derived relative to the Local
Standard of Rest (LSR), a fictitious object centred on the
Sun and orbiting the Galaxy at the local circular
velocity; the Sun orbits at a faster rate. The radial
velocity of an object in the Galactic plane is given by:
vR
= Θ cos α – Θ0 cos (90°–l)
= Θ cos α – Θ0 sin l .
where Θ is the circular velocity at distance R from the
Galactic centre and Θ0 is the circular velocity at R0, the
Sun’s distance from the Galactic centre.
By the Sine Law:
So cos   R0 sin l .
R
Therefore,
sin l sin 90    cos 


R
R0
R0
R0
vR 
sin l  0 sin l
R
  0 
 R0    sin l
 R R0 
 R0    0 sin l
since
0
 0
R0
and


R
Outside the Galactic plane the
radial velocity becomes:
vR  R0   0 sin l cos b .
The observed tangential velocity of the object relative to
the LSR is given by:
vT = Θ sin α – Θ0 cos l (where vT is positive in the
direction of Galactic rotation).
But R sin α = R0 cos l – d, where d is the distance to
the object.
R0
d
So
and
sin  
cos l 
R
R
 R0 

vT  R0 cos l  d   0   cos l
R
 R0 
  0 

 R0  
cos
l

d

R
 R R0 
 R0   0  cos l  d
These are the general equations
of Galactic rotation.
If Ω decreases with increasing distance from the Galactic
centre, then for any given value of l in the 1st (0° < l <
90°) and 4th (270° < l < 360°) quadrants, the maximum
value of Ω occurs at the tangent point along the line-ofsight, i.e. at Rmin = R0 sin l. In that case, d = R0 cos l, so:
Rmin = R0 cos (90° – l) = R0 sin l .
vR(max) = Θ(Rmin) – Θ0 sin l .
Approximations to the general formulae can be made for
relatively nearby objects, where d << R0, in which case:
 d 
  0  R  R0   
 dR  R0
and
But
1
2
R  R0 2  d 
 d 
   R  R0  
 dR  R0
 dR  R0
2



 d  
d  d 0  R  R0     0d .

 dR  R0 


d d    1 d 

 2 , so
 
dR dR  R  R dR R
1  d 
0
 d 
 dR   R  dR   R 2
R0
R0
0
0
And, for d << R0, R0  R ≈ d cos l .
So, for nearby objects in the Galactic plane vR becomes:
 1  d 
0 
vR  R0 R  R0  
  2  sin l
 R 0  dR  R0 R0 
 1  d 
0 
  R0  
  2  d sin l cos l
 R 0  dR  R0 R0 
  0  d  


  d sin l cos l
 R0  dR  R0 
  0  d  
 

  d sin 2l
 R0  dR  R0 
1
2
or vR = Ad sin 2l = Ad sin 2l cos2 b, outside the plane,
where:
 0  d  
1
A 2 

 
 R0  dR  R0 
is Oort’s constant A.
For the tangential velocity:
 1  d 
0 
vT  R0 R  R0  
  2  cos l   0d
 R 0  dR  R0 R0 
  0  d  
0
2


d
  d cos l 
R0
 R0  dR  R0 
  0  d  
0
 

d
  d 1  cos 2l  
R0
 R0  dR  R0 
1
2
  0  d  
 Ad cos 2l  

 d
 R0  dR  R0 
1
2
or vT = Ad cos 2l + Bd, where:
 0  d  
B 

 
 R0  dR  R0 
1
2
is Oort’s constant B.
Expectations from the
equations of motion are
that radial velocities
(solid line) and proper
motions(dashed line)
for nearby stars should
a double sine wave
variation with Galactic
longitude. They do.
The proper motion
relationship is a
modified version of
the vT relation:
vT  d  A cos 2l  B  so

A cos 2l  B 
ml 
4.74
In the 1st Galactic quadrant (0° < l < 90°) stars are
receding from the Sun.
In the 2nd Galactic quadrant (90° < l < 180°) stars are
approaching from the Sun.
In the 3rd Galactic quadrant (180° < l < 270°) stars are
receding from the Sun.
In the 4th Galactic
quadrant (270° < l < 360°)
stars are approaching
the Sun.
Note that:
 0  d  
A 

  and
 R0  dR  R0 
 d 
so A   12 R0 
 and
 dR  R0
1
2
Also:
0
0 
 A  B and
R0
 0  d  
B 

 
 R0  dR  R0 
1
2
B  A  0 .
 d 

   A  B
 dR  R0
So evaluation of Oort’s constants permits one to specify
the velocity gradient and local vorticity of local Galactic
rotation. It can also provide a solution for R0 if the local
rotational velocity can be found.
Can that be done?
Use of the equations for Galactic rotation is predicated
upon the establishment of an accurate value for the
Sun’s motion relative to the LSR. That is not an easy
chore because of the nature of stellar orbits in the
Galaxy, which are neither circular nor elliptical, but
more like a roseate pattern.
The general direction of the
Sun’s motion relative to
nearby stars is readily detected
in stellar proper motions, and
lies roughly towards
RA = 18h and Dec = +30°,
i.e. towards the constellation
of Hercules.
A typical orbit for a star in the Galaxy
can be pictured as epicyclic motion of
frequency κ superposed on circular
motion of frequency Ω. When κ = 2Ω
the orbit is an ellipse.
Since κ(R) ≠ 2Ω(R) in most cases,
the orbits are roseate, something
like what is produced by a
spirograph.
Cyclical motion perpendicular to
the Galactic plane also occurs.
The random motion of nearby
stars relative to each other
produces the observed velocity
dispersions for various stellar
groups.
Stars in the Galactic bulge
appear to exhibit no preferred
direction or orbital inclination,
so define a spheroidal
distribution.
The Local Standard of Rest:
In the gravitational field of the Galaxy, stars near the
Sun orbit the Galactic centre with velocities that are
close to the local circular velocity Θc. A star at the Sun’s
location that describes roughly a circular orbit about the
Galactic centre has velocity components:
(Π, Θ, Z) = (0, Θc, 0) km/s.
A velocity system centred on such a fictitious object is
used to define the LSR. That is, the LSR is defined by an
axial system aligned with the Π, Θ, and Z axes and with
an origin describing a circular orbit about the Galactic
centre with a velocity Θc.
Nearby stars have peculiar velocities relative to the LSR
described by:
u = Π – ΠLSR = Π ,
v = Θ – ΘLSR = Θ – Θc ,
w = Z – ZLSR = Z .
The peculiar velocity of the Sun is therefore given by:
(u, v, w) = (Π, Θ–Θc, Z) .
The velocity of any star with respect to the Sun has three
components:
i. a peculiar velocity relative to the star’s LSR,
ii. the peculiar velocity of the Sun with respect to the
Sun’s LSR, and
iii. the differential velocity of the LSR at the star with
respect to the solar LSR resulting from differential
Galactic rotation (usually negligible). Since (iii) is indeed
negligible for d ≤ 100 pc, the observed velocity of a star
relative to the Sun is given by the velocity vector (U*, V*,
W*), where:
U* = u* – u = Π* – Π ,
V* = v* – v = Θ* – Θ ,
W* = w* – w = Z* – Z .
For any particular group of stars belonging to the disk
and having nearly identical kinematic properties, one
can define a kinematic centroid of their velocities by:
1
u* 
N
N
u
*i
i 1
N
1
v* 
N
v
1
w* 
N
N
*i
i 1
w
*i
i 1
For disk stars not drifting either perpendicular to the
Galactic plane or in the direction of the Galactic centre,
it is reasonable to expect that:
u*  0 and
w*  0
However, <v*> ≠ 0, since a typical group of stars lags
behind the solar LSR. Stars chosen spectroscopically
include objects of various origins, unless the group is so
young that the stars have not had time to travel far from
their places of formation. The increasing density
gradient in the disk of the Galaxy towards the Galactic
centre implies that the majority of stars in the solar
neighbourhood originated at points lying on average
closer to the Galactic centre, i.e. most are currently near
apogalacticon. Since the apogalacticon velocity of stars
in elliptical orbits is less than the local circular velocity
c, any mix of elliptical orbits for nearby stars implies
the majority travel at less than c in the direction of the
Sun’s orbital motion. It follows that locally-defined
kinematic groups of stars should, in general, tend to lag
behind the LSR motion. Asymmetric drift is wellobserved in such groups, and must be taken into account
in any determination of the Sun’s LSR velocity.
From both qualitative and mathematical arguments the
asymmetric drift for any group of stars must depend
directly upon the nature of the orbits for the stars in the
group. For stars in nearly circular orbits no asymmetric
drift is expected, while for stars having a mix of very
eccentric orbits the asymmetric drift should be fairly
large. A group of stars having the latter properties
should also exhibit a fairly large dispersion in the
component of their orbital motions directed along the
line-of-sight to the Galactic centre, the  velocities,
whereas stars in strictly circular orbits have no such
component of their orbital motion. The correlation of
asymmetric drift with the dispersion in  velocities, 2,
proves to be a valuable tool for determining the exact
parameters for the Sun’s LSR velocity.
See the solution on the next slide, from work by the
instructor (Turner 2014, Can.J.Phys., 92, 959.)
u
v
w
= +11.1 ±0.5 km/s
= +4.4 ±0.6 km/s
= +7.3 ±0.2 km/s
corresponding to
S = 14.0 km/s towards
lLSR = 21°.5, bLSR = +31°.6.
Once the solar LSR velocity is established, it is possible
to establish likely distances to Galactic objects from their
radial velocities, i.e. using the equations of Galactic
equations of motion.
But all such efforts have used an incorrect solar LSR
velocity.
The rotation curve of the Galaxy is observed to be flat
like those of other disk galaxies, although perhaps not as
irregular as the solution obtained by Clemens. Note the
rigid body rotation near the Galactic centre.
Beginning in the 1970s a model for the propagation of
spiral features was proposed using a standing
logarithmic spiral density wave (“density wave model”),
but it rarely gives good agreement with what is observed
for our Galaxy’s spiral characteristics, except for some
features.
The spiral characteristics
of galaxies like M51 can
often be linked to
gravitational interaction
with a close companion
combined with
differential rotation in
the galactic plane. Could
that also be the case for
the Milky Way?
Methods of Establishing Distances:
i. Stellar parallaxes (trigonometric parallaxes, statistical
parallaxes, and secular parallaxes are used).
ii. Cluster parallaxes (see lab manual for 1st year).
iii. Spectroscopic parallaxes (calibrated from i and ii).
iv. Radial velocities and proper motions (approximate
only). Used for moving clusters.
v. Eclipsing binary light and radial velocity solutions.
The Galactic Centre:
All the evidence indicates the existence of a supermassive
object at the centre of the Galaxy, denoted as Sgr A*. It
is the source of X-rays and a cluster of interesting stars
orbiting it at high velocities. The large mass (~4106 M)
and compact nature of the object implies it is a black
hole, and appears to be typical of what is found at the
centres of many nearby massive galaxies.
Orbits of GC stars
according to Ghez et al.
(2004).
Example Problem. Find the mass of the Galaxy given the
local circular velocity of 251 km/s at the Sun’s location
roughly 8.5 kpc from the Galactic centre.
Solution.
Use Kepler’s 3rd Law in Newtonian fashion,
i.e. (MG + M) = a3/P2, for a in A.U. and P in years.
For an orbital speed of 251 km/s and orbital radius of 8.5
kpc the orbital period is:
2R
2  8500 pc  206265 AU/pc 1.496 108 km/AU
P

251 km/s
251 km/s  3.1558 107 s/yr
 2.0805 108 yr
The semi-major axis is:
a  8500 pc  206265 AU/pc  1.7533109 AU
So the mass of the Galaxy is:
MG 
1.753310 
2.0805 10 
9 3
8 2
M Sun  1.24511011 M Sun
So ~1011 M is derived for the mass of the Galaxy
internal to the Sun. If the orbital velocity curve is flat to
~16 kpc from the Galactic centre, then one can redo the
calculations to find that ~21011 M is derived for the
mass of the Galaxy internal to ~16 kpc from the centre.
Where did the extra ~1011 M come from, or is it proper
to apply Kepler’s 3rd Law in situations like this? Recall
that it applies to the case of a two-body situation only,
not to a multi-body situation.
Mass/Light Ratios, M/L:
The mass-luminosity relation varies roughly as L ~ M4
(M3 for cool stars), so the mass-to-light ratio should vary
as M/L ~ 1/M3 or 1/M2. The typical star near the Sun is a
cool M-dwarf with a mass of only 0.25 M or less,
implying a typical mass-to-light ratio for our Galaxy of
~16. Since most stars are probably less massive than
that, the actual mass-to-light ratio for the Galaxy could
be in excess of ~25 or so.
i. Stellar Parallaxes
Stellar parallax is the displacement in a star’s position in
the sky with respect to the stellar background arising
from the orbital motion of the Earth about the Sun.
Denoted by the angle π, it is defined to be the angle
subtended by 1 A.U. (the semi-major axis of the Earth’s
orbit) at the distance of the star. In practice one can
observe the annual displacement of a star resulting from
Earth’s orbit about the Sun as 2π.
In the skinny triangle approximation, 1 A.U. is the chord
length subtended at the star by the angle π, measured in
radians. In this case, the chord length ≈ the arc length
subtended at the star = dπ, where d is the distance to the
star. Since π < 1" for all stars, the equation can be written
as an equality, i.e. 1 A.U. = dπ, or:
In order to take advantage of trigonometric stellar
parallaxes measured in arcseconds, it is useful to define a
unit of distance corresponding to that angle. Thus, the
parsec is the distance to an object when 1 astronomical
unit (A.U.) subtends an angle of 1 arcsecond:
Since all stars should exhibit parallax, measured values
(trigonometric parallaxes) are of two types:
πrel = relative parallax, is the annual displacement of a
star measured relative to its nearby companions
πabs = absolute parallax, is the true parallax of a star, or
what is measured for it
In the past, all parallaxes were relative parallaxes, and
were adjusted to absolute via:
πabs = πrel + correction
The definition of
parallax and parsec =
distance at which one
Astronomical Unit
(A.U.) subtends an
angle of 1 arcsecond.
Note, by definition:
1 pc = 206265 A.U.
How parallax is
measured.
The complication of the
parallactic ellipse. In
practice all parallaxes
are measured using
only points near
maximum parallax
displacement.
The concept of relative
parallax is also
illustrated.
The estimated frequency of
an average 11th magnitude
star (upper) and an
average 16th magnitude
star (lower) as a function of
distance (visual magnitudes
dashed line, photographic
magnitudes solid line). 11th
magnitude stars peak for a
distance of ~250 pc,
corresponding to a
correction factor of 0".004.
16th magnitude stars peak
for a distance of ~800 pc,
corresponding to a
correction factor of
0".00125.
The distance to any star or object with a measured
absolute parallax is given by:
The relative uncertainty in distance is given by:
Typical corrections to absolute were +0".003 to +0".005
for old refractor parallaxes, but are roughly +0".001 for
more recent reflector parallaxes from the U.S. Naval
Observatory. Space-based parallaxes from the Hipparcos
mission are all absolute parallaxes; they were measured
relative to all other stars observed by the satellite. Their
uncertainties are less than 1 mas (milliarcsecond), i.e.
<0".001, although systematic errors of order 0".001 or
more are suspected in many cases. The Gaia mission will
measure parallaxes in similar fashion.
Example: What is the distance to the star Spica (α
Virginis), which has a measured parallax according to
Hipparcos of πabs = 12.44 ±0.86 mas?
Solution.
The distance to Spica is given by the parallax equation,
i.e.
The uncertainty is:
The distance to Spica is 80.39 ±5.56 parsecs.
ii. Cluster Parallaxes
The primary purpose of photometry is to obtain
information equivalent to spectroscopic data in a smaller
amount of observing time. It is also a highly efficient
method of studying variable stars. Stellar continua vary
with spectral type, and different photometric systems are
designed to sample selected portions of such continua
either for the equivalent of spectral classification or for
estimating third dimensions for stars, e.g. metallicity, etc.
Basically any photometric system must be capable of
determining a star’s spectral class, corrected for
interstellar extinction, without serious problems from
luminosity differences, or population effects. Systems are
classified on the basis of the widths of the wavelength
bands used to define them. Broad band systems use
passbands from 300 to 1000Å wide (e.g. the UBV system),
intermediate band systems 100 to 300Å wide (e.g. the
Strömgren uvby system), and narrow band systems less
than 100Å wide (e.g. Hβ photometry).
ii. Cluster Parallaxes
Typically such parallaxes (or distances) are derived from
photometry of stars in a cluster. Many photometric
systems exist, but a knowledge of the Johnson UBV
system helps to understand how other systems function.
For early-type stars, the Balmer discontinuity (λ3647)
and Paschen jump (λ8206) result from the opacity of H,
and are modified by
electron scattering in hot
O-type stars and H–
(negative H ion) opacity
in cool GK stars. The
Balmer discontinuity is
very sensitive to both Teff
and log g, so many
systems use filter sets to
isolate stellar continua on
either side of it.
Johnson’s UBV System.
The UBV system is designed to give magnitudes that are
similar to those on the old International photographic
and photovisual system, with a magnitude added on the
short wavelength side of the Balmer discontinuity in
order to give luminosity discrimination.
Filter λeff(nm) Δλ(nm)
U
365
68
B
440
98
V
550
89
U−B is sensitive to
gravity and reddening.
B−V is sensitive to
temperature and
reddening.
The diagram plotted
here is the standard
two-colour (or colourcolour) diagram for
the UBV system. Plotted
are the observed
intrinsic relation for
main-sequence stars of
the indicated spectral
types, the intrinsic locus
for black bodies
radiating at
temperatures of
4000K, 5000K, 6000K,
8000K, etc., and the
general effect on star
colours arising from the
effects of interstellar
reddening.
Intrinsic colours vary
according to both the
temperature and
luminosity of stars,
roughly as indicated
here.
Line blanketing affects
the observed colours of
late-type stars, low Z
stars exhibiting an
ultraviolet excess relative
to high Z stars. The
relationship is calibrated
relative to the Hyades
relation
at BV
= 0.60.
Blanketing lines vary in slope with spectral type,
which is why ultraviolet excesses are normalized
to BV = 0.60.
Examples of star clusters with uniformly reddened stars.
Examples of star clusters exhibiting small (left) and large
(right) amounts of differential reddening.
The reddening law can be described by:
where X is the slope and Y the curvature. Observations
indicate that Y = 0.02 ±0.01 while X varies with R locally.
See Turner 2014, Can.J.Phys., 92, 1696. In the Galactic
plane X can vary from 0.62 to 0.83, outside the plane
perhaps only X = 0.83 is applicable.
UBV intrinsic colours are presently tied to models for
non-rotating stars (left), but differential reddening still
dominates the observed colours (right).
The zero-age main sequence, as constructed from
overlapping the main sequences for various open clusters,
all tied to Hyades stars using the moving cluster method.
The present-day zero-age main sequence (ZAMS) for
solar metallicity stars.
Why it is important to correct for differential reddening
in open clusters: removal of random scatter from the
colour-magnitude diagram.
Differences between the core and halo regions of open
clusters. Note also the existence of main-sequence gaps.
Typical open cluster colour-magnitude diagrams
corrected for extinction. Note the pre-main sequence stars
in NGC 2264 (right) and the main-sequence gaps in NGC
1647 (left).
Many young clusters are also associated with very
beautiful H II regions.
ZAMS fitting can be done by matching a template ZAMS
(right) to the unreddened observations for a cluster (left).
The precision is typically no worse than ±0.05 (~2.5%).
Main-sequence fitting can be good to a precision of ±0.1
magnitude (±5%) in V0MV (or better) after dereddening.
An example of the importance of ZAMS fitting for open
clusters. The luminous and peculiar B2 Oe star P Cygni
belongs to an anonymous open cluster, so its reddening
and luminosity can be found using cluster stars.
The reddening (left) and ZAMS fit (right) for the P Cygni
cluster, an example of the usefulness of cluster studies.
Kinematic Method of Calibration ― Moving Clusters
Open clusters are ideally suited to the calibration of
stellar luminosities since they contain such a wide variety
of stars of different spectral types and luminosity classes.
The general method of using a calibrated zero-age mainsequence (ZAMS) to derive cluster distances is outlined
by Blaauw in Basic Astronomical Data. However, the
necessary zero-point calibration involves the independent
determination of the distance to a nearby cluster whose
unevolved main-sequence stars serve to establish the MV
versus (B–V)0 (or spectral type) relation over a limited
portion of the ZAMS. The distance to such a zero-point
cluster can be derived using the moving cluster method,
or one of its many variants.
The requirements for use of the moving cluster method
are a sizable motion of the cluster both across the line-ofsight and in the line-of-sight. The technique is most
frequently discussed for the case of the Hyades star
cluster, which is the standard cluster used for the
construction of the empirical stellar ZAMS. The
constellations Ursa Major and Scorpius also contain
moving clusters, and the Pleiades have attracted
considerable attention as a moving cluster. In general, all
stars in a moving cluster move together through space
with essentially identical space velocities (their peculiar
velocities are invariably much smaller by comparison).
Once the direction to the cluster convergent point
(divergent points are equivalent!) is established on the
celestial sphere, the geometry of the situation is
established. In particular, the angle  between the star’s
space velocity and radial velocity is fixed.
Motion of Hyades members.
The geometry
of proper motion.
Moving cluster
geometry.
The Hyades.
A star’s space velocity is given by: v2 = vT2 + vR2, and vT
= 4.74 μd. But, vR = v cos θ, and vT = v sin θ. Thus, the
distance to a star, d*, is given by:
Once the radial velocity, vR (in km/s), of a moving cluster
star, its proper motion μ (in "/yr), and angular distance
from the cluster convergent point, θ, are known, its
distance (in pc) can be obtained from the above equation.
The dispersion in radial velocity for stars in an open
cluster is typically quite small, no larger than ±1 to ±2
km/s. Therefore, for a cluster of stars of common
distance, the ratio (tan θ)/μ must remain constant across
the face of the cluster. It means that those stars lying
closest to the cluster convergent point have the smallest
proper motions, while those lying furthest from the
convergent point have the largest proper motions.
For nearby clusters that feature allows one to determine
the relative distances to stars in the cluster by comparing
the individual stellar proper motions, μ*, with the mean
cluster proper motion, μC, at that value of θ, i.e. using:
The technique has been
used extensively for
stars in the Hyades
cluster, which has a
line-of-sight distance
spread on the order of
10% or more of its
mean distance.
The moving cluster method can also be pictured in a
more general manner, as noted by Upton (AJ, 75, 1097,
1970). In this case, the motion of a cluster like the Hyades
away from the Sun results in an apparent decrease in the
cluster’s angular dimensions, even though its actual
dimensions are unchanged. By geometry and the
assumption that the cluster’s actual diameter, D, is
relatively constant, we have D ≈ rθ, where θ is the
cluster’s angular diameter. The cluster’s distance is
denoted r to avoid confusion with the derivative notation.
In practical terms:
The terms evaluated:
Since proper motion gradients are easier to measure
accurately than the location of the convergent point, the
method is somewhat superior to the convergent point
method. The cluster convergent point is not lost by the
method, since it is located by the points where the proper
motion gradients become zero. Upton noted, however,
that it was necessary to transform the α, δ motions of
cluster stars on the celestial sphere into their Cartesian
(flat surface) equivalents in order to obtain meaningful
proper motion gradients. For the Hyades cluster, he also
found it necessary to account for line-of-sight distance
spread.
A further modification to the general method can be
made using the original equations given previously,
namely:
i.e. given the mean proper motion of a moving cluster, one
can find its distance from the gradient in radial velocities
across the face of the cluster.
The last technique is very difficult to apply, since it
requires extremely accurate radial velocities for cluster
members that are not biased by the systematic effects of
binary companions. The method has been applied in a
very sophisticated fashion to the Hyades cluster by Gunn
et al. (AJ, 96, 198, 1988), with fairly good results. 1990
estimates for the distance modulus of the Hyades cluster
by all of the various methods used (including
trigonometric parallaxes) lie in the range 3.15 to 3.40, or
42.7 pc to 47.9 pc.
iii. Spectroscopic Parallaxes
These are normally done using tables of absolute
magnitude as a function of spectral type, e.g. Turner
ZAMS file.
Example: How distant is Spica (α Virginis), a B1 III-IV
star with apparent visual magnitude V = 0.91, given that
B1 III-IV stars typically have an absolute magnitude of
MV = –4.1 ±0.3 (Turner ZAMS)?
Solution.
The distance modulus for Spica is given by:
Thus:
The uncertainty is:
The spectroscopic parallax distance to Spica is 100.5
±13.9 parsecs. Note the small disagreement with the star’s
Hipparcos parallax distance of 80.39 ±5.56 parsecs.
iii. Statistical Parallaxes
Tables of absolute magnitude for stars as a function of
spectral type and luminosity class are constructed in a
variety of ways. It is not unusual for several different
techniques to be used in the compilation of one table,
including trigonometric parallax, cluster parallax, and,
finally, statistical parallax.
Recall the relation for tangential velocity,
for the proper motion μ in "/yr and the parallax π in
arcseconds. It can be rewritten as:
There is a statistical method of establishing the mean
parallax for a group of stars having common properties
by making use of the data for their positions in the sky,
their proper motions, and their radial velocities. The
resulting statistical parallax for such a group is:
where the angled brackets denote mean values.
In general, a randomly distributed group of stars should
have <vT> ≈ <vR>, i.e. one component of space velocity
should be similar to another. Thus,
That is a simplification of the true method. In practice,
the true situation is complicated by the Sun’s peculiar
motion relative to the group. Denote (upsilon)  = the mcomponent in the direction of the solar antapex (i.e. the mcomponent resulting from the Sun’s motion), (tau)  = the
m-component perpendicular to the direction of the solar
antapex (i.e. the m-component resulting from the star’s
peculiar velocity), and (lamda)  = the angular separation
of the star from the solar apex. Then:
is the secular parallax, the parallax inferred from the
Sun’s space motion, where vSun is the solar motion
relative to the local standard of rest (LSR), while:
is the statistical parallax, the parallax inferred from the
statistical properties of stellar space velocities.
Upsilon components apply when the Sun’s motion
dominates group random velocities; tau components
apply when group motions dominate. Both techniques
have been applied to B stars and RR Lyrae variables,
which are too distant for direct measurement of distance
by standard techniques. Both classes of object are also
relatively uncommon in terms of local space density, yet
luminous enough to be seen to large distances. Because of
general perturbations from smooth Galactic orbits
predicted in density-wave models of spiral structure, the
assumptions used for statistical and secular parallaxes
may not be strictly satisfied for many statistical samples
of stars.
An example for the B3 V stars, as published by Turner
2012, Odessa Astron. Publ., 25, 29 ― was a SMU lab.
Sample Questions
1. The star Delta Tauri is a member of the Hyades
moving cluster. It has a proper motion of 0".115/year, a
radial velocity of 38.6 km/s, and lies 29º.1 from the cluster
convergent point.
a. What is the star’s parallax?
b. What is the star’s distance in parsecs?
c. Another Hyades cluster member lies only 20º.0 from
the convergent point. What are its proper motion and
radial velocity?
Answer: The relationship applying to all stars in a moving
cluster is:
With the values given previously, the distance is found
from:
The star’s parallax is:
And its space velocity is:
The other star must share the same space velocity as
Delta Tauri, so its proper motion and radial velocity are:
2. The open cluster Bica 6, at l = 167º, has a radial
velocity of 57 km/s, or 48 km/s relative to the LSR. Its
distance from main-sequence fitting is 1.6 kpc. Is the
cluster’s motion in the Galaxy consistent with Galactic
rotation?
Answer. See
predictions at right.
The cluster should
have a LSR velocity
of −9 km/s if it
coincided with
Galactic rotation.
The cluster’s actual
LSR velocity of 48 km/s is therefore completely
inconsistent with Galactic rotation. Can you think of a
reason why?
3. Interstellar neutral hydrogen gas at l = 45º has a
radial velocity of +30 km/s relative to the LSR. What are
its distance from Earth and the Galactic centre if R0 = 8.5
kpc and θ0 = 251 km/s?
  
Answer. Recall:
vR  R0   0 sin l  R0   0  sin l
 R R0 
For a flat rotation curve, θ = θ0 = 251 km/s, so:
  0 
vR
30
   

 4.99 km/s/kpc

 R R0  R0 sin l 8.5 sin 45
1
1
 251

1
 

4
.
99
km/s/kpc

  0.1375328 kpc
R 251 km/s  8.5

1 kpc
R 
 7.2709884 kpc
0.1375328
For plane triangles the cosine law is:
Here:
a 2  b2  c2  2bc cos A
Angle A = 45º
Side b = 8.5 kpc = R0
Side a = 7.27 kpc = R
Side c = distance d to cloud
So:
Yielding:
7.272  8.52  d 2  28.5d cos 45
d 2  12.020815d  19.3971  0
Which is a quadratic equation with solution:
d
12.020815 
12.0208152  419.3971
2
 6.0104075  12 8.1799507  kpc, giving
d  1.92 kpc and 10.10 kpc