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5A-3 Astrophysics
Cosmology
Astrophysics booklet pages 54 to 65
August 21st, 2010
AQA A2 Specification
Lessons
1 to 5
Topics
1.4 Cosmology
Doppler effect
z = Δf / f = v / c and Δλ / λ = - v / cFor v « c applied to optical and radio frequencies.
Calculations on binary stars viewed in the plane of orbit, galaxies and quasars.
Hubble’s law
Red shift v = Hd
Simple interpretation as expansion of universe; estimation of age of universe, assuming H is constant.
Qualitative treatment of Big Bang theory including evidence from cosmological microwave background
radiation, and relative abundance of H and He.
Controversy concerning accelerating Universe and dark energy.(from section 1.3)
Quasars
Quasars as the most distant measurable objects.
Discovery as bright radio sources.
Quasars show large optical red shifts; estimation of distance.
The Doppler Effect
As the ambulance travels
forwards:
The sound wave in front is
compressed
- decreasing its wavelength
- increasing its frequency
A
A
B C
B C
D
D E
The sound wave behind is
stretched out:
- increasing its wavelength
- decreasing its frequency
ambulance
sound demo
The Doppler Effect also occurs with light and radio waves.
With light waves:
In front of the moving object
decreasing wavelength and increasing frequency
makes the light BLUER
Behind the moving object
increasing wavelength and decreasing frequency
makes the light REDER
Radial and tangential velocity
Consider a galaxy moving
relative to the Sun at velocity v.
This velocity can be split into two
components:
radial velocity, vr
vr = v cos θ
and tangential velocity, vt
vt = v sin θ
The Doppler Effect can be
used to measure radial
velocities.
v
vt
θ
vr
Doppler effect equations
When the radial speed of the object, v is
MUCH LESS than the speed of light, c.
change of frequency = object radial speed
source frequency
speed of light
Δf = v
f
c
Also:
change of wavelength = object radial speed
source wavelength
speed of light
Δλ = - v
λ
c
In either case, The Doppler Shift, z
is given by:
z = Δf = - Δλ = v
f
λ
c
Remember: v MUST BE << c
Question 1
Light of frequency
4.000 x 1014 Hz is emitted
from a galaxy that is
moving away from us at
600 kms-1. Calculate:
(a) the frequency shift
caused and,
(b) the observed frequency.
(a) v is << c
and so: Δf / f = v / c can be used
Δf = (v / c) x f
= (600 kms-1 / 300 000 kms-1)
x 4.000 x 1014 Hz
= 0.002 x 4.0 x 1014
frequency shift = 8.0 x 1011 Hz
c = 300 000 kms-1
(b) The galaxy is receding and so the
observed frequency is lower than the
source frequency.
source frequency = (4000 - 8) x 1011 Hz
= 3992 x 1011 Hz
observed frequency = 3.992 x 1014 Hz
Question 2
The Andromeda Galaxy has a
Doppler Shift of 0.000 5.
The observed wavelengths are
also smaller than the source
wavelengths.
(a) Calculate radial velocity of the
Andromeda Galaxy relative to the
Earth.
(b) What will be the observed
wavelength of red light if the
source was 700.00 nm?
c = 300 000
kms-1
(a) A Doppler Shift of 0.000 5 will mean
that v is << c
and so: Δλ / λ = - v / c = z can be used.
v=zxc
= 0.000 5 x 300 000 kms-1
speed = 150 kms-1
The wavelengths are decreased therefore the
Andromeda Galaxy is moving towards the Earth.
radial velocity = 150 kms-1 towards the Earth
(b) Δλ / λ = z
Δλ = z x λ
= 0.000 5 x 700.00 nm
= 0.35 nm
less than the source
observed wavelength = 699.65 nm
Question 3
A hydrogen radio source
emits waves of frequency
1420 MHz. A radio
telescope observes these
waves to have a frequency
of 1430 MHz.
Calculate radial velocity of
the source relative to the
Earth.
c = 300 000 kms-1
Δf = (1430 – 1420) MHz
= 10 MHz
Δf / f = z
z = 10 MHz / 1420 MHz
= 0.00704
A Doppler Shift of 0.00704 will mean
that v is << c
and so: z = v / c can be used
v=zxc
= 0.00704 x 300 000 kms-1
radial speed = 2 112 kms-1
The frequency is increased therefore the radio
source is moving towards the Earth.
radial velocity
= 2 112 kms-1 towards the Earth
Doppler shift summary
Doppler
shift, z
Source
moves
towards
observer
Source
moves away
from
observer
in frequency in wavelength
Δf
Δλ
f
λ
+ v
c
- v
c
- v
c
+ v
c
Colour shift
with light
BLUE
SHIFT
RED
SHIFT
Doppler shift in spectra
wavelength / nm
400
500
600
700
Hα
Stationary source
Hα
Receding source
RED SHIFT
Hα
Approaching source
BLUE SHIFT
Question
The hydrogen-alpha
spectra line of a distant
galaxy occurs at a
wavelength of 680 nm.
In the Sun this
wavelength is 656 nm.
Calculate the
recessional radial
velocity of the galaxy.
c = 300 000 kms-1
Δλ = (680 – 656) nm
= 24 nm
Δλ/ λ = z
z = 24 nm / 656 nm
= 0.0366
A Doppler Shift of 0.0366 will mean
that v is << c
and so: z = v / c can be used
v=zxc
= 0.0366 x 300 000 kms-1
= 10 980 kms-1
recessional speed = 11 000 kms-1
The Doppler effect with binary stars
A binary star system consists of two stars in mutual orbit
about each other.
The two stars are often too close together to be resolved by
a telescope as separate stars.
However, as they orbit each other they will be moving away
and towards the Earth.
The stars will produce both red and blue shifts which can
be measured.
This type of system is called a spectroscopic binary.
From the Doppler shift measurements the orbital speeds of
the stars can be found.
star B
star A
line of sight
SPECTRA
From star A
From star B
Blueshift
Red
shift
No shift
Blueshift
Red
shift
Combined
H-alpha line position
from a stationary source
In the example above, star A is more massive than star B.
This results in the orbital radius and speed of star A being less than star B.
Question
A spectral line of a certain
spectroscopic binary merges once
every 4 years and splits to a
maximum displacement of 0.036 nm
and 0.015 nm from its laboratory
wavelength of 656 nm.
Calculate:
(a) the orbital speed of each star and
(b) the radius of the larger orbit.
With the faster star:
v = 300 000 x (0.036 / 656)
faster star = 16.5 kms-1
c = 300 000 kms-1
The orbital speed of the faster star, v = 2πR / T
and so: R = Tv / 2π
= [(8 x 365 x 24 x 60 x 60s) x (16.5 kms-1)] / 2π
= [(252 288 000s) x (16 500 ms-1)] / 2π
= (4.162 x 1012 m) / 2π
= 6.63 x 1011 m
larger radius = 662 million km
The slower star has the smaller Δλ
= 0.015 nm
Δλ/ λ = v / c
becomes: v = c x (Δλ/ λ)
= 300 000 x (0.015 / 656)
slower star = 6.86 kms-1
The merging of the line occurs every half-period.
Therefore: T = 8 years
The faster star will have the greater orbital
radius, R
Hubble’s Law
In 1929 after plotting the redshift calculated speeds of
about thirty galaxies against their distances Hubble noticed
that a ‘rough’ straight line through the origin was obtained.
Hubble’s law states that the speed of recession of a
galaxy is proportional to the distance to the galaxy.
vαd
inserting a constant of proportionality:
v = Hd
H = the Hubble constant = 65 km s-1 Mpc-1
Notes:
(a) Mpc = megaparsec (= 3.26 million light years)
(b) Hubble’s law does not work with some nearby galaxies
(which are approaching us!)
Question 1
Calculate or state the speeds expected for galaxies at
distances: (a) 1 Mpc; (b) 10 Mpc & (c) 1 billion light years.
1 Mpc = 3.26 million light years
(a) From the definition of the Hubble constant, H:
A galaxy at distance 1 Mpc will have a speed of 65 km s-1
(b) At 10 Mpc the speed will be 650 km s-1
(c) 1 billion light years = (1000 / 3.26) Mpc
= 306.7 Mpc
therefore speed = 306.7 x 65
speed = 19 900 km s-1
Question 2
Calculate the distance, in light years, to a galaxy if its
recessional velocity is 10% of the speed of light.
c = 300 000 kms-1 & 1 pc = 3.26 lyr
v = Hd
therefore:
d=v/H
= (0.1 x 300 000 kms-1) / (65 km s-1 Mpc-1)
= 30 000 / 65
= 461.5 Mpc
= (461.5 x 3.26) Mlyr
= 1 504 Mlyr
= 1.5 billion light years
The expansion of the Universe
Hubble’s law tells us that distant galaxies are
receding from us and that the further they are
away the faster they are moving away from us.
Hence the Universe is expanding.
Balloon model of the
expanding Universe
The space between the
galaxies expands and so the
galaxies grow further apart
Estimating the age of the Universe
Hubble’s constant tells us that the speed of a galaxy increases by
65 kms-1 for every Mpc or 3.26 million light years.
For a galaxy to be receding at the speed of light its distance would
be:
(300 000 kms-1) / (65 kms-1 Mpc-1)
= 4 615 Mpc
= 4 615 x 3.26 million light years
= 15 billion light years
Galaxies cannot travel this fast.
Therefore light cannot have been travelling for this length of time.
Therefore the Universe must be less than 15 billion years old.
The current greatest length of time observed is about 13.5 billion
years.
As can be seen in the above calculation, the maximum size of the
Universe, D is given by:
D=c/H
But distance = speed x time
For a galaxy travelling at speed c for the age of the Universe T.
D=cT
Therefore: c / H = c T
1/H= T
Hence the maximum age of the Universe is given by:
T=1
H
Note: All of the above assumes that the Hubble constant has not
changed during the lifetime of the Universe.
Question 1
The unit of 1 / H must be one of time. Calculate the
value of 1 / H in seconds. 1 Mpc = 3.1 x 1022 m
H = 65 kms-1 Mpc-1
Therefore 1 / H = 0.0154 Mpc s km-1
But 1 Mpc = 3.1 x 1022 m = 3.1 x 1019 km
Hence:
1 / H = 0.0154 x 3.1 x 1019 km s km-1
= 4.77 x 1017 s (15 000 million years)
Question 2
In 2009 measurements from the Hubble Space Telescope yielded a
Hubble constant of 74.2 ± 3.6 km s-1 Mpc-1. What is the maximum age
of the Universe, in years, based on this measurement?
1 Mpc = 3.1 x 1019 km
T=1/H
Maximum T will come from a minimum value of H.
that is: 74.2 - 3.6
= 70.6 km s-1 Mpc-1
Therefore: T = 1 / 70.6
= 0.0142 Mpc s km-1
But 1 Mpc = 3.1 x 1019 km
Hence: 1 / H = 0.0142 x 3.1 x 1019 km s km-1
= 4.40 x 1017 s
= (4.40 x 1017 s) / (365 x 24 x 60 x 60) years
Maximum age = 14 000 million years
The Big Bang theory
• Hubble’s law supports the idea that the
Universe started in some form of massive
explosion.
• This idea is called the ‘Big Bang theory’.
• This theory was not generally accepted
until 1965 before which an alternative
explanation for the expansion of the
Universe called the ‘Steady State theory’
was dominant.
Evidence for the Big Bang theory
1. Cosmic Microwave Background (CMB) radiation
All of space emits thermal radiation with a maximum intensity
wavelength corresponding to a temperature of 2.7K.
It is radiation created in the Big Bang that has been travelling ever
since the Universe became transparent (about 377 000 years after the
Big Bang).
The expansion of the Universe has gradually increased the
wavelengths of this radiation so that it now occurs in the microwave
region of the electromagnetic spectrum.
This radiation was first detected by Penzias and Wilson in 1965.
Expanding Universe
The microwave detector used
by Penzias and Wilson
Cosmic Microwave Background radiation.
CMB map of the whole Universe obtained
by the COBE satellite in 1989
2. Relative abundance of hydrogen to helium
Using the Big Bang model it is possible to calculate the
ratio of concentration of hydrogen to helium-4 by mass.
This depends on the ratio of photons to baryons, which
itself can be calculated independently from the detailed
structure of Cosmic Microwave Background fluctuations.
The ratio by mass predicted is about 4:1.
The measured ratio of abundance is 3:1 which is roughly in
agreement.
Other ratios such as hydrogen to Lithium-7 or Helium-3
also give roughly similar results.
Revision Question
Calculate the wavelength and frequency associated with a
thermal temperature of 2.7K.
Wein’s displacement law: λmax T = 0.0029 mK
λmax x 2.7K = 0.0029 mK
λmax = 0.0029 / 2.7
wavelength = 0.0011 m (1.1 mm)
c=fλ
becomes: f = c / λ
= (300 000 000 m/s) / (0.0011m)
frequency = 2.7 x 1011 Hz (270 GHz)
Revision of supernovae as standard candles
Type 1a supernovae have a known peak luminosity
allowing them to be used as ‘standard candles’.
At their peak all of these supernovae have an absolute
magnitude, M of -19.3 ± 0.03.
By noting their apparent peak magnitude, m such
supernovae can be used to determine this distances to
galaxies using the equation:
m – M = 5 log (d / 10)
Question
In a distant galaxy a Type 1a
supernova is observed to
have an apparent magnitude
of + 21 and a recessional
speed (from red-shift
measurement) of
60 000 kms-1.
Calculate the distance to this
galaxy (a) from the magnitude
measurement (b) using
Hubble’s law.
Assume Type 1a supernovae
have an absolute magnitude
of – 19.
Comment on your answers
(a) m – M = 5 log (d / 10)
rearranged:
log (d / 10) = [(m – M) / 5]
= [(21 – (-19) / 5]
= 40 / 5 = 8.0
antiloging: = 100 x 106 = d / 10
Distance to galaxy = 1000 Mpc
(b) Hubble’s law: v = Hd
therefore: d = v / H
= (60 000 kms-1) / (65 km s-1 Mpc-1)
= 923 Mpc
Distance to galaxy = 920 Mpc
(c) The magnitude measurement gives a
greater distance measurement than the
one obtained from red-shift measurement.
Dark energy
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In 1998 astronomers discovered that very distant type 1a
supernovae were further away than expected. Measurement of their
red shifts (used to measure their velocities) and the use of Hubble’s
Law to obtain their distances shows that these supernovae are
fainter than expected.
These mesurements indicate that the Universe is expanding faster
now than when the supernovae exploded as the light has had to
travel further to reach us than expected by a constant rate of
expansion (assummed to be the case with Hubble’s Law).
These and further measurements have led astronomers to conclude
that the expansion of the Universe has been accelerating for about
the past 5000 million years.
Before this discovery, the expansion of the Universe was expected to
be decelerating due to the attractive gravitational forces of galaxies
on each other.
Therefore there appears to be some unknown repulsive force acting
which is releasing some hidden ‘potential’ energy which is currently
known as dark energy.
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•
The nature of dark energy is unclear. It is thought to be a form of
background energy present throughout space and time.
It is more prominent than gravity at large distances as gravity
becomes weaker with distance (inverse square law) whereas the
force associated with dark energy is thought to remain constant.
Current theories suggest that it makes up 70% of the total energy of
the Universe.
The idea of dark energy leads to the speculation that in the future
the force associated with dark energy will ultimately tear apart all
gravitationally bound structures, including galaxies and solar
systems, and eventually overcome the electrical and nuclear forces
to tear apart atoms themselves, ending the universe in a "Big Rip".
Dark energy controversy
It has been noted that if Newton’s law of
gravitation:
F = GMm/r2
became: F = GMm/r
at large, intergalactic distances,
then the acceleration of the expansion of the
universe no longer requires the existence of Dark
Energy.
Other alternative ideas for dark energy have come
from string theory, brane cosmology and the
holographic principle.
Quasars
‘Quasar’ stand for ‘quasi-stellar radio source’.
A quasar is a compact region in the centre of a
massive galaxy surrounding a central
supermassive black hole.
Its size is 10-10,000 times the Schwarzschild
radius of the black hole.
Artist’s impression of a quasar
The quasar is powered by an accretion disc
around the black hole.
Quasars were first identified as being high
redshift sources of electromagnetic energy,
including radio waves and visible light, that
were point-like, similar to stars, rather than
extended sources similar to galaxies.
Infra-red image
of a quasar.
Quasars are among the oldest and most distant
objects in the Universe.
A quasar is characterised by:
–
–
–
its very powerful light output, much greater than
the most massive of stars
its relatively small size, not much larger than a
star
a large red shift indicating its distance is between
5 and 10 billion light years away.
The first quasar discovered, 3C 273 produced
strong radio emissions. However, not all quasars
produce such emissions.
Question
Light from a certain quasar
was found to contain a
spectral line that had been
red-shifted by 80 nm from its
normal wavelength of 486 nm.
Calculate the recessional
speed of this quasar.
c = 300 000 kms-1
z = Δλ / λ
= 80 nm / 486 nm
= 0.164
A Doppler Shift of 0.164 will mean
that v reasonably smaller than c
and so: z = v / c can be used to a
reasonable accuracy
v=zxc
= 0.164 x 300 000 kms-1
recessional speed
= 49 300 kms-1
Internet Links
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Red Shift - eChalk
Expanding Universe - 7stones
Expanding Universe - eChalk
Age of the Universe - eChalk
Big Bang Time Machine - School Science
Doppler Effect - Sound with microphone
Doppler Effect - Iona
Doppler Effect - Explore Science
Doppler Effect (Quick Sound Demo) - Iona
Doppler effect - NTNU
Doppler effect - ambulance - Fendt
Doppler Shift with light - eChalk
Doppler Shift with sound - eChalk
Doppler Shift with sound effects- eChalk
Core Notes from Student Guide pages 54 to 65
1.
2.
3.
4.
5.
Draw a diagram to explain what
is meant by the ‘Doppler effect’
State the various equations
relating the wavelength and
frequency change observed with
the motion of the radiating body.
What must be true for these
equations to be used with
reasonable accuracy?
Explain how the Doppler effect
can be used to determine the
speeds of the stars that form a
spectroscopic binary.
State Hubble’s law.
Explain how Hubble’s law leads
on to the idea of the expanding
Universe and the Big Bang
theory.
6.
Show how the Hubble constant
can yield an estimate for the
age of the Universe.
7.
What evidence is there for the
Big Bang theory?
8.
What is dark energy? Explain
why it is needed to account for
the observations of some type
1a supernovae.
9.
What is a Quasar?
10. What are the main
characteristics of quasars?
Notes from the Student Guide pages 54 to 58
3.1 The Doppler Effect
1.
2.
3.
4.
5.
6.
Draw a diagram to explain what is meant by the ‘Doppler effect’
State the various equations relating the wavelength and frequency
change observed with the motion of the radiating body. What must
be true for these equations to be used with reasonable accuracy?
Explain how the Doppler effect can be used to determine the
speeds of the stars that form a spectroscopic binary.
Repeat the worked example on page 56 but this time for a star
whose spectral line is displaced by – 0.025 nm.
Repeat the worked example on page 57 but this time for a binary
whose lines merge every 0.80 years with the same spectral
displacements from a laboratory wavelength of 520 nm. Do not
repeat the error made in the worked example!
Try the summary questions on page 58
Notes from the Student Guide pages 59 to 63
3.2 Hubble’s Law & Beyond
1.
2.
3.
4.
5.
6.
7.
8.
State Hubble’s law.
Explain how Hubble’s law leads on to the idea of the expanding
Universe and the Big Bang theory.
Show how the Hubble constant can yield an estimate for the age
of the Universe.
What evidence is there for the Big Bang theory?
What is dark energy? Explain why it is needed to account for the
observations of some type 1a supernovae.
Repeat the worked example on page 60 but this time for a galaxy
whose spectral line is observed as 400.0 nm.
If H was 75 kms-1 Mpc-1, what is the maximum age of the
Universe?
Try the summary questions on page 63
Notes from the Student Guide pages 64 & 65
3.3 Quasars
1. What is a Quasar?
2. What are the main characteristics of
quasars?
3. Try the summary questions on page 65