Astrophysics
 18
lecture introduction
-10 lectures on cosmology
-8 lectures on stellar evolution
one guest lecture by Matthew Young on Pulsars
 Power point slides plus one lecture from a PDF
 Two minor, one major assignment: see handout
 Slides will be put on web
 Text Carroll and Ostlie: Modern Astrophysics
 Contact me whenever necessary: ext 2736, mob
0409687703,dgb@physics, rm4-67, basement lab, Gingin 95757591
This course includes material from lectures at many major universites and
institutes including Chicago, Fermilab, Stanford, Sheffield. Authors include R
Kolb, Mohr, W.Hu, V Kudryavtsev,
Course Outline
See handout
Cosmology: 10 lectures
Stellar Evolution: 8 lectures
Excursion: Wed 16 March: barbeque, cosmology and
astronomy field night.
Time: leave UWA 4.30pm: bus and car pool. Return by
11pm.
Major Assignment: Dark Energy, Black hole binary systems
and Intermediate mass black holes.
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Major Assignment
The major assignment asks you to write an investigation on one of
three topics based on recent research
a) dark energy and the missing mass
b) intermediate mass black holes. ( eg 1000 solar mass near
galactic centre)
c) stellar mass black hole binary systems (predicted, none
discovered, many expected in gravitational wave signals)
The investigation should be based on recent discovery papers. In
your investigation you must demonstrate that you have read and
understood at least 3 research letters. Show how they relate to
each other. Use Nature, Astrophysical Journal Letters, Science
and arXiv Astro-ph or gr-qc preprints.
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Looking into the past
•Telescopes are time machines
• Looking into the past we see a
universe that is
–Hotter: - thermal background
radiation rising in temperature
–Denser: -galaxies are closer
together
–Expanding:-Everything is
receeding
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History of the Universe
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Cosmology and Dark Matter
• First 2 lectures
Hubble law
Critical density
Density parameter
Mass to light ratio
Dark Matter in solar system
Dark Matter in galaxies
Dark Matter in clusters of galaxies and superclusters
Conclusions
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Introduction
First hints for ‘dark matter’ (1844):
1. It was noticed that planet Uranus had moved from its
calculated position by 2 arc minutes.
2. F. W. Bessell found the strange motion of the star
Sirius.
By 1846 the planet Neptune was discovered (no longer a
dark matter).
In 1862 the faint companion to Sirius (Sirius B - a white
dwarf) was discovered.
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Redshift
wavelength at the point of observation
l obs - lem
z =
l em
wavelength at the
point of emission.
In terms of the velocity of the receding object red shift is given by:
1+ v/c
1+z =
1- v/c
v
z≈
c
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The Hubble Law
1 parsec = 3.25 ly.
Stellar parallax from
earth orbit.
QuickTime™ and a
GIF decompressor
are needed to see this picture.
Cepheid variables
standard candles
In 1929, based on the observation that the universe is expanding it was
further realised, by Hubble, that the expansion velocity v is proportional
to distance away from the observer (Earth) r:
v = H0 r
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Hubble’s Law
H0 is the Hubble constant - the rate of expansion at the present time.
v H0
r
z≈ =
c c
The precise value of H0 was disputed for many years.
Today cosmologists agree H0 ~ 70 km s-1 Mpc-1
-1
-1
H0= h . 100 km s Mpc
h = 0.70 ± 0.05
Hubble thought H0 was 540 km s-1 Mpc-1
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Critical density
Mass m
r
d
contributing mass
This requires the famous Newtonian
result:
a) a particle inside a spherical mass
distribution feels no gravitational
force.
b) For a particle outside a spherical
mass distribution the gravitational
force is as if all the matter were
concentrated at central point.
e.g. 1) The force exerted on the Earth by
the moon depends on the mass of
the moon and not on its density
profile.
2) The gravitational acceleration of
the earth falls to zero as you
approach the core.
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Critical Density
rc
Consider the motion of a galaxy of mass m at the edge of a
spherical region of mass M and radius r :
Kinetic energy
T = mv2/2
U = -GMm/r
Potential energy at the edge of a sphere
Total energy
E = T + U = mv2/2 - GMm/r
The mass of the sphere can be calculated from its volume and
mean density
M = 4p r3 r / 3
The critical density of the Universe is the density which gives E = 0
3v2
rc =
8p G r 2
2
3H0
rc =
8p G
critical density
From known H0 we can compute the value of the critical density:
rc(t0)
= 1.88 h2 . 10-26 kg m-3
(i.e. small)
6 H atoms per m3
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Density Parameter
W0
The density parameter W0 is the ratio of the true density of the Universe at
the present time to the critical density:
r0
W0 =
rc
density parameter
Open Universe:
0 < W0 < 1
r0 < rc
Flat Universe:
W0 = 1
r0 = rc
Closed Universe:
W0 > 1
r0 > rc
Note that we can use the density parameter to quantify components of the
density due to particular types of material in terms of the ratio to the
critical density, i.e. Wrad, Wmatter, Whalo, etc.
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Fate of the Universe
W<1
empty
W=0
open
W=1
flat
W>1
closed
The Friedmann Equations
Open Universe:
will expand forever
Flat Universe:
will expand forever
(but the expansion
rate slows to zero at
infinite size)
Closed Universe:
will end in a Big
Crunch
The fate of the Universe, as well as many other things, depend on
the density (density parameter).
Can we measure it?
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Age Size and Lifetime of Closed Universes
For closed solutions the size of the Universe will reach a maximum
q0 = 1
empty
decelleration
parameter qo
open
flat
closed
2c
amax =
H0
t L= 2p
H0
2
t 0=
3H0
also we can calculate the lifespan of the Universe - the time from birth to
recollapse, e.g. for q0 = 1.
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A crude estimate of the density
A crude estimate comes form considering the typical mass of a
galaxy ~ 1011 MSun, and typical galaxy separation ~ 1 Mpc. Check for
yourself that this gives a density close to the critical density.
W~ 1
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Mass to light ratio
Mass to light ratio can help us to find the density or W0.
There are stars which are intrinsically faint, such as white dwarfs,
brown dwarfs. There are also dead stars, such as neutron stars
and black holes.
Important to distinguish between objects which are intrinsically
dim and those which are dim because they are very distant.
We can define the mass to luminosity ratio for a given system
(galaxy, cluster of galaxies, any part of galaxy) relative to the Sun.
We define the mass to light ratio as:
M
L
= h
MSun
LSun
where h = 1 for the Sun
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MSun
LSun
= 0.51
g
erg/s
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Mass to light ratio
Characterise the average density in various regions of the Universe in
terms of mass-to-light ratio. and contribution to the density parameter.
Note M/L is proportional to h.
Two examples: i) h <<1 :a system is composed of massive ,
young and luminous main sequence stars; i) h >>1 : a
system with old white dwarfs and hidden (dark) matter.
Measurement of M/L depends on location:
Solar neighbourhood: count up the luminosity of all stars etc and determine
masses from orbital motions.
In galaxies measure total galaxy luminosity and use rotation curves or virial
theorem (see later) to estimate total mass.
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Dark Matter in the Galaxy
Consider motion
of the stars
perpendicular to
the Galactic
plane.
Assume this
motion is
independent of
the conventional
circular motion
around the
Galactic centre.
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Dark Matter near the Sun - the
Oort limit
The velocity in z-direction vz decreases as z increases
due to the gravitational attraction to the Galactic plane.
It is impossible to measure vz or gz (the gravitational
force per unit mass).
But assuming the big number of oscillations made
around the plane and mapping the distributions of stars
away from the plane, it is possible to estimate the
average gravitational force.
In general we have:
gz = g0 (z / z0)
where g0 and z0 are measurable constants of
acceleration and length.
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The Oort limit
The acceleration due to gravitational attraction is thus:
dgz / dz = - 4 p G r
where r is the average density of gravitating matter.
Thus near the Sun (Oort limit - 1932, 1965):
r = g0 / 4 p G z0
Oort estimated this as 0.2 MSun/pc3. Recent estimates give the total
density of 0.15 MSun/pc3 (0.3 GeV/m3) and 0.08 MSun/pc3 for stars and gas
only.
Hence about 1/2 or 1/3 is missing matter. But this is at a very specific
point (near the Sun).
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Density in stars and other luminous matter
We can estimate the mass contribution from stars in galaxies. We add up
the mass of stars, making use of the known relationship between stellar
luminosity, temperature and mass. Within the optical radius of galaxies
this yields a non-dynamical estimate for the mass density.
M100 spiral
W stars ~ 0.005
This provides a lower limit for the
baryonic density. It can be extended
slightly by integrating over the total
background
luminosity
of
the
Universe, but it still yields a value no
more than:
W visible baryons = 0.01
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Density estimated from galactic dynamics
Rotation curves of spiral galaxies
The first real evidence for substantial
dark matter came in 1970 with
Freeman’s observation of the rotation
curves of Galactic halos. He showed
that the 21 cm line of neutral hydrogen
did not show the expected Keplerian
decline beyond the optical radii of
these galaxies.
What would we expect if all the mass
of a galaxy were accounted for by the
visible mass?
The galaxy "M51". Messier 51 is also known as NGC 5194 and
sometimes called the Whirlpool galaxy. The distance to M51 is
about 9 Mpc
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Rotation curves of spiral galaxies
Assume that stars, gas and dust move in circular orbits around galaxy. At
large distances the gravitation field would be as if all the mass were
concentrated in the centre
Centripetal force is balanced by gravity:
2
circ
GM(r)
GM(r)
vcirc =
=gr
=
2
r
r
r
Keplerian decline v ~ r -1/2 is not observed
v
Doppler shift can be due to several motions:
i) motion of the whole galaxy away/towards us;
ii) random motions of the clouds;
iii) rotation of the galaxy.
This can be separated to find the rotation curve: we can measure v0 + vcirc
and v0 - vcirc . This applies to galaxies seen edge on or at an angle.
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Rotation curves of spiral galaxies
In almost all galaxies the velocity is found to be constant with radius.
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Rotation curves of spiral galaxies
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Rotation curve of the Milky Way
300
Milky Way
10-20
10-21
10-22
mean 10-23
density 10-24
(g cm-3) 10-25
10-26
10-27
10-28
10-29
/kms-1
200
HI (layer)
HI (Tangent)
HI (Petrovkaya)
HII (regions)
PNs
100
LUMINOUS MATTER
actual density (Galactic disk)
mean spherical density
0
0
~0.3 GeV cm-3
DARK MATTER
5
10
R/kpc
15
20
mean spherical density
Sun
0
~30 kpc
1.1023
LMC
2.1023
distance from galactic centre (cm)
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Rotation curves of spiral galaxies
From the observations we can try to model the density distribution:
The easiest model is to assume the galaxy is spherical (we don’t need
to assume that the hidden mass is distributed like the visible mass):
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Rotation curves of spiral galaxies
We can use:
gr =
GM(r )
r2
where M(r) is the mass within radius r.
In this case we can determine the mass distribution uniquely. The
solution is not much different for the more realistic case of a flattened
spheroid. The best fit mass density distribution is:
r(r) =
2
0 0
rr
(r2 + r02 )
ro, ro
are constants
This yields a typical total mass to light ratio
in the halo (lower limit is for visible part of
galaxy):
A similar result is obtained when considering
orbiting pairs of galaxies.
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hhalo ~ 10 -100
W halo ≈ 0.1 (maximal)
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Density estimates for elliptical galaxies
To determine the mass of elliptical galaxies we need to use the virial theorem
because there is not much rotation. Alternatively we can use X-rays in halos or
pairs of galaxies. All this gives:
hhalo ~ 10 -100
W halo ≈ 0.1
M87 Elliptical
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(maximal)
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Density of clusters from bulk motions
Estimates by the virial theorem
In any system of gravitating bodies changes in size are determined by the
balance between gravitational attraction and the motions of the bodies.
Example: Coma
For orbit, mv2/r=GmM/r2 Hence twice the
kinetic energy T is equal to the negative of the
gravitation potential energy V. (Correction
factor a)
2T + V = 0
1
2
T= M<v >
2
Cluster mass
a = 0.5 - 2.0
V =-
aGM
2
R
Mean galaxy velocity
2
<v >R
M=
aG
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hclusters ~ 200
W clusters ≈ 0.2
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Problems with clusters
• We have to be sure that the galaxies are actually in the cluster (redshift).
• Exclude large fluctuactions (fast moving galaxies).
• Carefully treat close clusters moving near each other.
• Account for total velocity (not just vz).
• Be sure that this is a cluster (not a random coincidence) and that the
cluster is not contracting or flying apart.
• Take into account cosmological evolution (galaxy formation).
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Estimates from x-ray observations
The properties of the hot gas that emit x-rays can be used to determine the
mass and density profile of the dark matter, even though they may not
themselves have the same density profile.
Example: Coma
The temperature maps can be
used to determine the mass
needed to prevent the hot
gas and
galaxies from
escaping the clusters. One of
the best examples is the
analysis
for
the
Coma
cluster.
The result is:
hclusters ≈ 300
W cluster ≈ 0.3
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Estimates from x-ray observations
Chandra X-ray
Observatory
images (left, Xrays from hot gas)
and Hubble Space
Telescope images
(right, massive
central regions
bend light from
distant galaxies)
of the giant galaxy
clusters Abell 2390
and MS2137.32353. The clusters
are located 2.5
and 3.1 billion light
years from Earth,
respectively.
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Gravitational lensing
A relatively new
technique of
measuring the Dark
Matter is to use the
gravitational
deflection of light
rays by the cluster.
This distorts the
image of
background
objects giving arclike features which
are magnified
images of distant
galaxies behind the
cluster.
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Density estimated from gravitational lensing
hclusters ≈ 300
W cluster ≈ 0.3
a distant galaxy lensed by a nearer galaxy cluster
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Density estimated from gravitational lensing
Map of a galaxy cluster using gravitational lensing
showing the Dark Matter distribution
Tyson et al.
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Density from supercluster dynamics
The total mass of superclusters is obtained from deep redshift galaxy
surveys, again using virial techniques. Comprehensive surveys of infra-red
and other galaxies have gone out to distances in excess of 200 Mpc. From
large-scale velocities, it is possible using linear theory to estimate the
homogeneous mass density.
HST deep field
hsuperclusters = 800±500
W supercluster= 0.8  0.5
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Density from supercluster dynamics
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Density from theory - structure and inflation
There are no current working models of structure formation that do not
require dark matter with:
W DM ≥ 0.3
Inflation, the model that may explain the Universe in its early stages when
it undergoes rapid expansion, predicts that the Universe is flat. In the
simplest form this tells us:
W0 = 1
However, we could have a lower density of matter if we assume the
presence of dark energy, which is favoured by recent observations of
distant type Ia supernovae (cosmological constant).
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Conclusions
1
Theoretical Expectation
Large Scale
Structure
Clusters of Galaxies
Non-baryonic
DM problem
0.1
W
Galaxy: All Mass
Nucleosynthesis
Baryon Limits
0.01
Galaxy: Visible Mass
0.001
0.0001
Baryonic
DM problem
(i)
There is a missing mass
(dark matter) in the
Universe;
(ii) It is seen at all scales
from
galaxies
to
superclusters;
(iii) It is also predicted by
the theory (simulations
of
the
structure
evolution, inflation).
Solar Neighbourhood
0.01
100
1
10000
Scale (Mpc)
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