Download Advancing Physics A2

Name ………………………………………………………
Advancing Physics A2
John Mascall
The King’s School, Ely
Chapter 12 Our place in the universe
(Parts 1 and 3)
Student Notes
August 2009
Assessable learning outcomes for all of Chapter 12
Candidates should demonstrate evidence of:
1. knowledge and understanding of phenomena, concepts and relationships by describing:
(i) the use of radar-type measurements to determine distances within the solar system; how
distance is measured and defined in units of time, assuming the relativistic principle of the
invariance of the speed of light;
(ii) effect of relativistic time dilation using the relativistic factor
  1 1 v2 / c2 ;
(iii) the measurement of relative velocities by radar observation;
(simple arguments using two successive pulses are sufficient);
(iv) evidence of a ‘hot big bang’ origin of the universe from:
• cosmological
red-shifts (Hubble’s law);
• cosmological
microwave background;
2. scientific communication and comprehension of the language and representations of physics, by
sketching and interpreting:
(i) logarithmic scales of magnitudes of quantities: distance, size, mass, energy, power,
brightness;
3. quantitative and mathematical skills, knowledge and understanding by making calculations and
estimates involving:
(i) distances and ages of astronomical objects;
(ii) distances and relative velocities from radar-type measurements.
A Revision Checklist for Chapter 12 can be found on the Advancing Physics
CD-ROM.
Page | 2
Ch 12.1 Observing the Universe
Learning outcomes

Astronomical distances in the Solar System can be measured by radar

Distances to nearby stars can be found by parallax

Some larger distances are estimated by the apparent brightness of ‘standard
candles’, e.g. Cepheid variables and Type 1a supernovae

The cosmological distance scale is still subject to uncertainty

For v much less than c, velocities of astronomical objects can be established by the
Doppler shift with
λ v

λ c

Masses of astronomical objects can be found from the velocities of objects orbiting
them at known distances or in a known time.

Spectra of distant objects over a wide range of wavelengths provide knowledge of
their chemical composition
Looking out on the Universe
We start with a discussion around what you already know about cosmology and ask the
question, 'And how do you know?'.
Activity 10E Experiment 'What do you know about cosmology?'
Many people have a great interest in cosmology – and already know a lot about it. But
what they know may be disorganised - and self contradictory. This activity will help you
find out how much you already know about this topic.
We follow this by taking a look at images of the Universe at a wide range of wavelengths:
This first computer
image (Display Material
20S Computer Screen
'The Milky Way in
infrared') is given as an
example. All can be
found on the CD-ROM)
and on the PowerPoint
‘Looking in on the
Universe’.
Display Material 20S
Display Material 40S
Display Material 50S
Display Material 60S
Display Material 70S
Computer Screen 'The Milky Way in infrared'
Computer Screen 'Eta Carinae in visible light'
Computer Screen 'An ultraviolet image of the Sun'
Computer Screen 'Images of supernova remnant Cassiopeia A'
Computer Screen 'X-ray and gamma ray images'
Page | 3
The following OHTs help to summarise what we know about astronomical distances.
Display Material 10O
OHT 'Distances in light travel time'
Page | 4
Distances in light travel-time
1027 m
1000 million
light-years
1 million
light-years
limit of
observable
Universe
Hubble deep field
1024 m
distance to
Andromeda
galaxy
1021 m
size of Milky Way
galaxy
10 18 m
1 light-year
10 15 m
next-nearest
star
Alpha Centauri
Sun to Pluto
1 light-hour
10
12
m
Sun to Earth
Sp eed o f ligh t
1 light-second
109 m
106 m
1 lightmicrosecond
103 m
Earth to M oon
300 000 kilometres per second
300 km per millisecond
300 m per microsecond
300 mm per nanosecond
Ligh t travel-tim e
One hour by jet
plane
ten m inute walk
1
1
1
1
light- second = 300 000 km
light- millisecond = 300 km
light- micros econd = 300 m
light- nanosecond = 300 mm
Tim e
1 year = 31.5 million seconds
1 lightnanosecond
1m
L ig ht-year
1 light-year = 9.4  10 15 m
16
= 10 m approximately
Page | 5
Display Material 30O
OHT 'The ladder of astronomical distances'
Ladder of astronomical distances
10 10
Red-shift
Assume that the speed of recession
as measured by wavelength shift is
proportional to distance
Brightest galaxy
Assume that brightest galaxies
in clusters are all equally bright
10 9
Super nova
Type 1a supernovae all have
the same absoute brightness
Coma cluster
10 8
Virgo cluster of galaxies
10 7
Tully-F isher
F aster rotating galaxies have
greater mass and are brighter
M31 Andromeda
10 6
Magellanic clouds
Blue supergiants
Assume that the brightest
star in a galaxy is as bright
as the brightest in another
10 5
10 4
Cepheid variables
T hese very bright pulsing stars
can be seen at great distances .
T he bigger thay are the brighter
they s hine and the slower they
pulsate.
10 3
10 2
10
1
Colour-luminosity
The hotter a star the brighter its light, and
the brighter it shines. If the type of star can
be identified there is a known relationship
between colour and brightness. Distance
then found comparing actual with apparent
brightness
Parallax
Shift in apparent position as Earth moves
in orbit round Sun. Rec ently improv ed
by using satelite Hippar cos: now
overlaps Cepheid scale.
Baseline
all distances based on measurement of solar system, previously using parallax, today using
radar
Page | 6
Above are some techniques for measuring distance, arranged to show how they overlap,
allowing comparisons between the techniques to be made.
The following experiments give you some idea of how to find out about objects that cannot
be reached. They provide useful background information but are not essential.
Activity 20E Experiment 'Range finding and parallax'
In the activity you will learn about trigonometric parallax and how the distances to nearby
stars can be measured.
Activity 30E Experiment 'A homemade reflection spectroscope'
You will use this to observe the Sun’s spectrum and to look for absorption lines – missing
lines in the spectrum caused by absorption of certain wavelengths by the Sun’s
atmosphere. The PowerPoint presentation called ‘Spectra’ shows the three main types of
spectra together with an example of the Sun’s spectrum.
Activity 40E Experiment 'Brightness and distance'
Using the intensity of starlight to measure the distance to the stars is often difficult.
You have to assume the brightness of the star, and then use the way the brightness of the
star varies as you get further away, perhaps assuming a 1/r2 variation, to produce a
distance.
Activity 50E Experiment 'Summer Sun remembered’
In this activity you compare energies incident on the face which allows you to make an
estimate of the distance of the Sun.
Notes:
Page | 7
Radar ranging and Doppler shift
By using a single pulse of radio waves we can measure how far a distant object is away
provided it is not too far. You should appreciate why there is no possibility of using this
method to measure the distance to the nearest star beyond the Sun, which is 4.2 lightyears away.
If we send two pulses of radio waves a known time interval apart we can measure the
relative velocity of a moving object. Again, this method will not work if the object is too far
away, but it will work for asteroids passing close to the Earth.
Display Material 90O
OHT 'Velocities from radar ranging'
Radar ranging
re lativ e
velocity v
relative
ve locity v
asteroid
as teroid
firs t radar
pulse out
spe ed of light
8
–1
= 3  10 m s
firs t radar
puls e returns
s ec ond radar s econd radar
pu lse o ut
pulse returns
fir st p u lse
seco nd p ulse
time betwe en first and
sec ond pulses 10 0 s
first pulse
out
first puls e
returns
sec ond pulse
out
0.2 s
sec ond puls e
re turns
0.22 s
100 s
distance out and bac k
= 0.2 light-secon ds
distance out and bac k
= 0.22 light-secon ds
distanc e of asteroid
= 0.1 light-s econd s
= 30 000 k m
distanc e of asteroid
= 0.11 light-sec onds
= 33 000 km
time taken
= 100 s
increase in dis tance
3000 k m = 0.01 light-se conds
relative veloc ity v =
3000 km
100 s
–1
v = 30 k m s
4
v/c = 0. 01 s/100 s = 10 –
The relative velocity of an asteroid is calculated by measuring two outand-back radar pulses
Page | 8
For objects that are a very long way away (distant stars and galaxies) the signal would
take too long to return and would be too weak to detect. In this case we must make use of
the Doppler effect. Atoms in the distant stars emit light at particular wavelengths. If we can
identify the elements from which the light is emitted it is possible to measure these
wavelengths accurately on Earth. If the stars or galaxies are moving away from us we will
detect an increase in wavelength when the light reaches Earth. This is known as a redshift.
Display Material 95O
OHT 'Non-relativistic Doppler shift'
Doppler shift
source and receiver both at rest
c = speed of light
f = frequency
T = time for 1 cycle
receiver
source
w avefronts
wavelength
=
c
= cT
f
v
+  cT+ vT
=
=1 +
cT
c

 v
=
c

wavelength
larger by 
+  = cT + vT
receiver
source
moving
and receiver
at rest
velocity v
source
travels
distance vT
in each
cycle
The wavelength increases when the source travels away from the receiver
We can illustrate the Doppler effect with the sound from a Formula 1 car as it races past or
with a Doppler ball swung around the head!
The Doppler effect is particularly well illustrated in context in the IOP Schools’ Lecture
presentation by Pete Edwards called ‘Gravity, Gas and Stardust’.
Page | 9
Some or all of the following activities may be used:
Activity 110P Presentation 'Changes in velocity and rotation'
Also uses File 50L Launchable File 'Doppler seen and heard'
Many objects in the Universe spin on their axis or orbit around a more massive body.
Typically this will result in components of their velocity, as seen from Earth, varying.
Sometimes the radiating atoms on the object will be, on average, moving towards the
detector, and sometimes away from the detector. So the frequency of the detected waves
will vary. Often this will be the only evidence available. Exactly this observation has been
used to claim the first direct sighting of a planet, 55 million light-years away, in December
1999. Here you look at a model of this kind of remote sensing of velocity.
Activity 120D
Demonstration ‘ Doppler shift using microwaves’
In this experiment, a moving metal sheet is used to reflect microwaves and, because the
sheet is moving, the frequency of the reflected waves is shifted. The frequency shift
depends on the speed and direction of the movement.
Demo Activity 90S Software Based 'The space police'
Using File 30L
Launchable File 'Radar ranging model'
Key point:
∆ = v
 c
where v  c
See pages 68-69 of the text.
Question
A spectral line resulting from a transition within hydrogen atoms associated with a quasar
is found have wavelength of 475.0 nm when detected on Earth. The same transition from
hydrogen in the laboratory produces radiation of wavelength 410.2 nm.
(a) Explain why this implies that the quasar is moving away from us.
(b) Use the Doppler formula to calculate the velocity of the quasar.
Notes:
Page | 10
Doppler shifts (used for measuring speeds) have been crucial to detecting rotations of
galaxies and other objects. Here the rotation is used to calculate the mass of a candidate
black hole.
The argument for measuring the mass of black hole is developed on pp 71-72 of the text.
Display Material 190O
OHT 'Measuring black holes'
Measuring the mass of a black hole
–1
velocity 209 km s away from us
acc eleration g =
com panion s tar
v2
r
ac cele ration meas ures the
grav itational field of the black hole
acce leratio n g
cand idate
black hole
field g =
m as s M
radi us r
GM
r2
=
GM
r2
v2
r
2
v eloc ity 209 km s
–1
mas s from v and r M = v r
G
towards us
Example: mass o f a cand idate black h ole V404 Cygn us
V404 Cy gnus emi ts X-ray s perhaps due to m atter from an ordinary star falling into a m ass ive black hole c ompanion
w hich it orbits . Dopple r s hifts of light from the ordinary s tar show its ve locity varying by plus or min us 209 k m s–1 over
a period of 6.5 d ays .
speed v in o rbit
from D oppler measurements , ass uming orbital plane
is in the lin e of s ight
radiu s r o f orb it
from time of orbit and speed of star:
1
star travels 2r in 6.5 days at 209 km s –
acceleratio n a to ward s b lack h ole
acc eleration =
v2
r
=
9
r = 18.7  10 m
(209  10 3 m s–1)2
2
a = 2 .34 m s–
18.7  10 m
9
s am e quantity
1
g = 2.34 N k g –
gravitatio nal field g of black
gravitational field = acc eleration
ma ss M o f b lack ho le
from grav itational invers e squarefield g =
GM
r
–1
2
M=
9
gr
2.34 Nkg  (18.7  10 m )
=
G
6.67  10 – 11 Nm 2 kg – 2
m ass of Sun = 2  10
30
kg
1
v = 2 09 k m s–
2
2
M = 1.2  10
31
kg
mas s M = 6 times m ass of Sun
The mass of a black hole can be calculated using the velocity of a star
orbiting it
Page | 11
Evidence of red and blue shifts from the same galaxy is given below:
Display Material 130S
Computer Screen 'Doppler shifts from part of a galaxy'
Images of the central gas disc
of galaxy M87, and the spectra
of emission line gas from that
central disc, showing both red
shifts and blue shifts. These
show that one side of this gas
disc is approaching us whilst
the other side is receding. The
disc seems to be rotating at a
speed of about 550 km s-1 .
There is no evidence that the
whole galaxy is rotating.
Page | 12
Ch 12.2 Special relativity
Learning outcomes

Motion is relative: 'being 'at rest' means only 'moving with me'.

The speed of light c is constant, the same relative to all objects, however they
move.

The relativistic Doppler shift approximates closely to the non-relativistic value
1 + v / c for v substantially less than c, but increases rapidly as v approaches c.

The time dilation factor γ is very close to 1 for v substantially less than c but
increases rapidly as v approaches c.
We start with a discussion of what is involved in ‘thinking relatively’. In Question 50C
Comprehension ‘Thinking relatively’ you have to learn to describe events seen from
different points of view, and work out what changes and what stays the same. These
questions provide simple exercises in thinking in this way and require no knowledge of the
results of the theory of relativity. Answers to each question are provided on a PowerPoint
to support class discussion.
The following limerick taken from the student text encapsulates the ideas met so far.
When Einstein was travelling to Spain,
He drove the conductor insane:
“It may be a while,”
He would ask with a smile,
“But when does Madrid reach this train?”
We have seen in Chapter 11 that the law of conservation of momentum applies regardless
of the speed of a moving observation platform. Travelling at a different constant velocity
relative to another object does not change the laws of physics.
We must accept another essential idea about light and that is that its speed is the same
regardless of the speed of the observer. That speed is now defined as 299 792 458 ms -1.
There is good experimental evidence to support the constancy of the speed of light and
this is backed up by a coherent set of theoretical arguments. Some evidence is outlined on
pages 76-77 of the student text.
The two postulates that underpin special relativity are as follows:
Postulate 1
The laws of physics are the same for all observers moving at a constant velocity.
Postulate 2
The speed of light is the same for all observers. It does not depend on the speed
of the source or the speed of the observer.
Page | 13
In Section 12.1 we discussed a radar method to measure the distance and speed of an
asteroid. We assumed that the radar pulses travelled at a constant speed we now know to
be the same constant speed for all electromagnetic waves when travelling in a vacuum.
One pulse gave us a single distance measurement and two pulses gave us a change of
distance in a known time.
In the arguments below we use space-time diagrams to represent radar measurements.
Display Material 100O
OHT 'Space-time diagrams'
Space time diagram s
tim e t/s 10
pulse
back
4s
back
tim e t/s 10
radar pulse
out and back
radar pulse
out and bac k
pulse
back
8
x /c = 4 s
3s
back
6
8
6
x /c = 3 s
4s
out
4
radar puls e
out and back
2
pulse out
your w orldline
trav elling in time
3s
out
worldline
of objec t
not
moving
relative to
you goes
vertic ally
0
4
2
pulse out
sloping w orldline
of object m oving
relative
to you
0
2
4
distance x/c
6
2
4
dis tance x /c
6
D istance is m easured by reflecting a radar pulse then measuring the time sent and the tim e received
back . T he clock us ed travels w ith you and the radar equipm ent (at rest).
As sum ptions:
speed c is constant s o reflection oc curs halfw ay through the out-and-back time
speed c is not affected by the m otion of the distant objec t
Radar measures distances using light-travel times
The constant speed of light is assumed
The key features of a space-time diagram are given on page 77 of the student text. You
should make sure you are understand these diagrams before you move on.
Questions
1
Why does the worldline for the moving object in the right-hand panel have a greater
slope than that for the radar pulse?
…………………………………………………………………………………………………
…………………………………………………………………………………………………
Page | 14
2
(a) Using information given on the right-hand panel, calculate how far away from
you the moving object is at the moment the radar pulse reaches it. Give your
answer in both light-seconds and metres.
…………………………………………………………………………………………………
…………………………………………………………………………………………………
(b) How far away is the moving object at the moment the radar pulse returns to
you? Choose the most appropriate unit for your answer.
…………………………………………………………………………………………………
…………………………………………………………………………………………………
Measuring speed
Display Material 110O
OHT 'Two-way radar speed measurement'
Two-way radar speed m easurm ent (simplified)
time t/s
10
worldline of
observer
pulse 2 back t2
distance at moment of reflection
worldline of
moving object
from pulse travel time
1
= c 2 (t2 – t1 )
8
from object travel time
1
= v 2 (t2 + t1 )
6
pulse is reflected
5
4
v t2 – t 1
=
c t2 + t 1
1
(t – t1 )
2 2
1
(t + t1 )
2 2
2
pulse 2 out t1
t2 1 + v/c
=
t1 1 – v/c
pulse 1 out and back instantly
0
3 ls
distance x/c
pulse takes 6 s to go and return
distance = 3 light-s
pulse sent at t = 2 s must have
been reflected at t = 5 s
6 ls
t2 = 8 s t2
=4
t1 = 2 s t1
v 8s–2 s 3
=
=
c 8s+2s 5
Speed is measured by comparing the interval between returning pulses with the interval at which they were
sent
Note that, for simplicity, the object moves past the observer at t = 0 and the first radar
pulse is sent out at that time. Clearly it takes negligible time for this pulse to be returned
after being sent out. The second pulse can then be used to find out how far the object
travels in a certain time.
The panel shows that the distance at the moment of reflection can be expressed
as c(t2 – t1)/2 and as v(t2 + t1)/2.
Page | 15
Questions
1
Show that equating these two distances gives v/c = (t2 – t1)/(t2 + t1) as shown.
…………………………………………………………………………………………………
…………………………………………………………………………………………………
…………………………………………………………………………………………………
…………………………………………………………………………………………………
2
Re-arrange the equation v/c = (t2 – t1)/(t2 + t1) to show that t2/t1= (1 + v/c)/(1-v/c) as
given.
…………………………………………………………………………………………………
…………………………………………………………………………………………………
…………………………………………………………………………………………………
…………………………………………………………………………………………………
Doppler effect
Remembering that the first pulse went out and returned at t = 0, you should be able to use
OHT 110O to convince yourself that:
t1 = interval between pulses sent out and
t2 = interval between pulses coming back.
It therefore follows that
interval between pulses coming back = ∆t back = 1 + v/c
interval between pulses sent out
∆t out
1- v/c
Question
Show that ∆t back / ∆t out when v = 0.1 c is 1.22.
…………………………………………………………………………………………………………
…………………………………………………………………………………………………………
…………………………………………………………………………………………………………
You have shown that ∆t back is 1.22 larger than ∆t out. This stretching of the interval between
a pair of pulses is the Doppler effect occurring twice.
Firstly, the pulses are being sent to an object moving away from the original source of the
pulses. We say that this introduces a Doppler stretching factor k.
Secondly, the pulses are returned to the observer from the object that now acts as a
moving source. This introduces another Doppler stretching factor k.
Page | 16
It follows that the interval between the pulses when they return is given by
∆t back = k × k × ∆t out = k2 ∆t out
Question
Calculate the value of k in the numerical example given above.
…………………………………………………………………………………………………………
…………………………………………………………………………………………………………
…………………………………………………………………………………………………………
We can now establish that
∆t back = k2 = 1 + v/c
∆t out
1- v/c
The arguments are summarised below.
OHT 'Doppler shift – two-way and one-way'
Display Material 120O
Doppler shift – two-way and one-way
time t 10
B
A
Let the one-way Doppler shift = k
Pulses sent from A at intervals t
will arrive at B at intervals k t out
8
pulses return to
observer at intervals
t back = k  ktout
The same shift k must apply to pulses
sent from B to A
Therefore:
pulses arrive back at A at intervals
t back = k  kt out
6
pulses arrive at
moving object at
larger intervals
ktout
4
pulses sent out
at intervals t out
The two-way Doppler shift = k 2
Two-way Doppler shift
1 + v/c
k2 =
1 – v/c
2
One-way Doppler shift
1 + v/c
k=
1 – v/c
0
2
4
distance x/c
6
The Doppler shift k is the observed quantity that measures the speed of a remote object
We are normally interested in the one-way Doppler effect for which
k
1 v / c
1 v / c
Page | 17
We met k as the stretching factor for the time between a pair of pulses when the object is
moving away. This applies to a one-way trip only. The same idea applies to the stretching
of wavelength when source and observer are moving apart. This follows since the
wavelength is proportional to the time for one cycle.
The relativity equations we have just developed now replace the previous equations given
for the Doppler effect (see page 68 of the student text).
From relativity we have
k
received
1 v / c

sent
1 v / c
where λreceived = λ + ∆λ
λsent
λ
The previous expression for the Doppler effect (no reference to relativity) was
λ + ∆λ = 1 + v/c
λ
Whilst these equations look very different, the panel below shows that they give very
similar values when v « c.
The panel below shows that the relativistic Doppler shift can be written as
λ + ∆λ = γ(1 + v/c)
λ
where

1
1 v2 / c2
Questions
1
Show that γ is only 1.005 when v = 0.1 c.
…………………………………………………………………………………………………………
…………………………………………………………………………………………………………
…………………………………………………………………………………………………………
2
Explain why the simpler expression for the Doppler effect is adequate when speeds
are lower than 0.1 c.
…………………………………………………………………………………………………………
…………………………………………………………………………………………………………
…………………………………………………………………………………………………………
Page | 18
Display Material 125O
OHT 'Relativistic effects'
Relativistic effects
Relativistic Doppler shift is:
Relativistic and non-relativistic Doppler shifts
7
+
=

1+v/c
1–v/c
6
relativ istic Do ppler s hift
 (1 + v /c )
5
4
3
identic al for v < < c
2
multiply by
1+v/c
=1
1+v/c
1
non- relativ istic Doppler shift 1 + v /c
0
0
0.1
0.2
0 .3
0.4
0.5
v/c
0.6
0.7
0.8
0.9
1.0
0.9
1.0
giving
+
1+v/c
=

(1–v/c)(1+v/c)
Relativistic factor 
4
 =
1
2
2
(1 – v /c )
 inc reases towards
infin ity as v approac hes c
3
which is
 is alw ay s > 1
+
1+v/c
=

(1–v 2/c2)
2
 ~~ 1 for all v << c
1
0
0
0.1
0.2
0.3
0.4
0.5
v/c
0.6
0.7
0.8
it is neater to write this as
+ (1+v/c)
=

with
1
=
1–v 2/c 2
At low speeds relativistic and non-relativistic results are the same.
At speeds near c the relativistic factor  increases rapidly
The following activities are designed to support the work on the relativistic Doppler effect.
Activity 100S Software-based ‘The relativistic Doppler effect’
In this exercise you will explore the relativistic Doppler effect that gives red shifts and blue
shifts in the spectra of stars and galaxies according to whether they are moving away from
us or towards us. You will be able to turn off the effect of special relativity and see what
happens. In this way you will learn some surprising things about time.
Activity 50S Software-based ‘Investigating the relativistic Doppler shift’
The theory of relativity gives the relativistic Doppler shift k as
k
1  v / c 
1  v / c 
Here you study the way that k varies with the ratio v / c. You can compare the values with
the non-relativistic approximation 1 + v / c for a moving source.
Page | 19
Time dilation and the relativistic factor γ
In the simple Doppler shift equation, the factor 1 + v/c comes about because the source is
moving away from the observer stretching out the light waves. The term γ in the relativistic
equation γ (1 + v/c) arises because clocks in the systems moving relative to each other
cannot agree. A clock moving relative to you runs slower by a factor γ compared with a
clock moving at your speed. This is known as time dilation.
In the panel below a clock moves sideways to avoid any classic Doppler shift. Any
difference between the clocks can only be due to time dilation. The clock used is known as
a light-clock consisting of a pair of mirrors between which pulses of light bounce back and
forth.
Display Material 13O
OHT 'The light clock'
The light clock
sitting beside the clock
clock travelling past you at speed v
Pythagoras’ theorem:
(c)2 = (c)2 – x 2
 2 = t 2 – (x/c) 2
d = c
c
c
c
x = vt
x = vt
mirrors d = c apart
time out and back
(1 tick) = 2
clock records
wristwatch time 
You see the light take a longer path
but the speed is still c
So the time t is longer.
The moving clock ticks more slowly.
time dilation
t = t
=
with
substitute distance
x = vt
2 = t 2 (1 – v2/c2 )
gives
1
1 – v 2 /c 2
t=

1 – v 2 /c 2
Time dilation is a consequence of the constant speed of light
In this case, the clock going past takes a longer time t for each tick than its own
‘wristwatch’ time τ.
The relation between the two is
t =γτ
where

1
1 v2 / c2
The time dilation effect has been confirmed experimentally using muons (see Question
120S). Examples of time dilation in practice are given on page 82 of the student text.
Page | 20
The following activities are designed to support the work on time dilation.
Activity 110S Software-based ‘The light clock’
When you run the clock you should be able to understand how it works so that you can
see that clocks which move past you must tick slower than any you carry with you showing
your own 'wrist watch time'.
Activity 60S Software-based ‘Investigating the time dilation factor γ’
Here you study the way the time dilation factor  varies with the ratio v / c. You can look for
ranges of values in which it is undetectable, and ranges of values for which it is large.
Page | 21
Ch 12.3 Was there a ‘Big Bang’?

Red shifts of distant galaxies give evidence of the expansion of the Universe. A red
shift
z


corresponds to an expansion in scale of
RNOW
 1 z
RTHEN



Current estimates of the expansion time-scale of the Universe put it at about
14 ± 2 Gy.
Evidence that the Universe has evolved from an initial uniform, hot dense state
comes from the existence of the cosmic microwave background.
There are still fundamental problems in explaining the major features of the
Universe.
An expanding Universe
We present just two key pieces of evidence for the expansion and origin of the Universe.
The two key ideas are the cosmological red-shift and the cosmic microwave background,
as evidence of the expansion of the Universe from a ‘hot big bang’ origin. No reference is
made to the relative proportions of the elements in the early Universe.
It is important to appreciate that at large red shifts, we are not looking at ‘recession
velocities’ in any simple sense. The space of the whole Universe is expanding, stretching
wavelengths with it. Thus Doppler shifts, which do indicate relative velocities for nearer
objects, now indicate an expansion of space.
Hubble established the relation between red-shift and distance but the value for the
Hubble constant has made rather dramatic changes in its value over time. A larger value
of H0 would mean a more rapidly expanding Universe.
v = H0r
(v = recession velocity in kms-1 and r = distance in Mpc where 1 Mpc = 3.09  1019 m)
Notes:
Page | 22
Some experimental evidence of the relation between the red-shift and the distance to five
galaxies is shown below (see p 84).
Display Material 180O
OHT 'Red shifts of galactic spectra’
The photographs are of the brightest galaxies in successively more distant
clusters, together with observed red shifts in the light from these galaxies. The
two dark absorption lines are due to calcium – they are called the H and K lines.
Page | 23
There are real and fundamental difficulties involved in the measurement of distances on
this scale. As a consequence the ‘accepted’ value of the Hubble constant has changed
many times.
Display Material 140O OHT 'How the accepted value of the Hubble constant has changed'
How estimates of the Hubble time have changed
+
Hubble 1929
600
+
+ +
2
+
400
+
+
200
+
age of Earth
4.6 Gyr
+
+
100
+
+
++
over 300 m easurements in
disputed range 50–100
5
+
+
10
20
de Vaucouleurs
current best value
13.7 G yr
0
1920
1940
1960
1980
2000
Sandage
year
Cosmic convergence – the graph shows the evolution of the Hubble constant
measurements from the 1920s until the present day.
Over 300 measurements of the Hubble constant, H0, have been published since 1975.
Recent results are converging towards a value of 65 km s-1 Mpc-1.
Page | 24
Display Material 160O
OHT 'Hubble's law and the age of the Universe'
Hubble’s law and the age of the Universe
Hubble found that the further away a galaxy is, the
larger its redshift.
He interpreted this to mean that distant galaxies are
receding from us.
For a galaxy a distance d from us, Hubble wrote
v=Hd
where v is the speed of a galaxy away from us and
H is a constant called Hubble’s constant.
Run the Universe backwards in time...
...distant galaxies are further away but are moving
faster...
...in the past galaxies must have been closer
together...
...even further back, all the matter and space in the
Universe was concentrated at a single point.
A galaxy distance d from us takes a time t = d/v to reach us in a reversed
Universe. From Hubble’s law:
d d
1
t=v=
=
Hd H
This time is independent of d and v and tells us how long ago the Universe was
a single point - this is the age of the Universe.
Strictly, in a reversed Universe, the galaxies accelerate as they fall together so
that the ‘Hubble time’, 1/H, gives an upper limit for the age of the Universe.
Page | 25
Display Material 200O
OHT 'The age of the Universe'
The ‘age of the Universe’
+
+
+
+
+
+
+
+
distance to galaxy
speed of recession is directly proportional to distance v = H0 r
1
= Hubble time
H0
H0 is the Hubble constant
units of H0 are
speed
1
=
time
distance
If speed v were constant, then
units of
1
= time
H0
1
r
= = time since galaxies were close together
H0 v
If we assume that the speed v was constant, the Hubble time would represent the time
since galaxies were close together. This time represents the age of the Universe.
Exercise
(a) Show that a Hubble constant of 65 km s-1 Mpc-1 leads to an estimate of 15 Gyr for the
age of the Universe. [Hint: take great care with the units]
…………………………………………………………………………………………………………
…………………………………………………………………………………………………………
…………………………………………………………………………………………………………
…………………………………………………………………………………………………………
…………………………………………………………………………………………………………
…………………………………………………………………………………………………………
Page | 26
(b) Suggest why the Universe must be younger than this time.
…………………………………………………………………………………………………………
…………………………………………………………………………………………………………
…………………………………………………………………………………………………………
…………………………………………………………………………………………………………
…………………………………………………………………………………………………………
…………………………………………………………………………………………………………
Notes:
Page | 27
Display Material 150O
OHT 'The history of the Universe'
These timelines in two different forms show the history of the Universe on a logarithmic
scale, illustrating important events.
Times since the Big Bang / s
101
107
1013
108
1014
103
109
1015
104
1010
1016
105
1011
1017
10
2
He nuclei formed.
Proportions of matter
in the Universe fixed
Neutral atoms formed.
Photons travel freely
through the Universe
Galaxies formed
Solar system formed
Life arises on Earth
Extinction of the dinosaurs
The present
106
1012
1018
Page | 28
A spiral timeline showing the history of the Universe on a logarithmic scale.
The history of the Universe
M oving inwards
along the tim e
spiral, eac h point is
one tenth of the
age of the previous
one, m easured in
s econds .
T he c urve spirals
endles sly inw ards
to the Big Bang.
Neutral a tom s form ed.
Photons trav el freely
through the U nivers e
10 14
10 6
10 15
10 13
10 7
10 5
10 16 1 0 8
10 4 10 12
10 2
10 9
10 3
Helium nuc lei form ed
Galaxies formed
C urrent theories
do not go nearer
the Big Bang than
10 –36 s
1 0 17
Solar s ys te m form ed
10 11
10 10
10 18
N ow
Extinction of the dinosaurs
Page | 29
Display Material 170O
OHT 'Relativity and the expanding Universe'
The expanding Universe according to general relativity
According to general
relativity, the Big Bang
was not an explosion of
matter into empty space.
Both space and matter came into existence together about 14 billion years ago.
Imagine a balloon with dots drawn on
it to represent galaxies, and slowly
blow it up.
As the balloon grows, the
space between the
galaxies grows. The
galaxies do not move
within the surface.
In the real Universe - unlike in the balloon model - the
galaxies themselves do not grow: their gravity holds
them together.
General relativity pictures the expansion of spacetime as if it were an expanding balloon.
This animation helps to bring alive the idea of an expansion of space itself.
Activity 130S Software Based 'The cosmological red shift'
Using File 70L Launchable File 'The space expands' which is launched from the CD ROM
\components\Flash\120079f1.exe
Page | 30
The cos mological red-shift...
Think of an electromagnetic wave drawn on the balloon, travelling from one galaxy to another.
A light wave travels from galaxy 1 to galaxy 2. T he galaxies are a distance de mitte d apart
when the wave is emitted.
The Universe expands.

em itted
d
em itted
Space is stretched and the wave with it.
rec eiv ed
d rece ived
When the light is received, the galaxies are a distance d receiv ed apart.
Wavelengths are red shifted because spacetime stretches as the light travels through it.
The expansion of space is related to the cosmological red shift.
Page | 31
Cosmological red-shift
Light travels from one galaxy to another, as the Universe
expands
light
em itted
Space stretching
Wavelength stretching
em itted
em itted
R emitted
 observed
R emitted
R observed
R observed
=
R emitted
observed
em itted
The Universe expands...the photons travel
 observed
light observed
R observed
R emitted
R observed
R emitted
R observed
R observed
R emitted
=
em itted + 
em itted
=1+

emitted
observed = emitted + 

=Z
emitted
=1+z
Question
Radio galaxy 3C324 has a red shift z = 1.12. Calculate by what percentage the universe
has expanded since the light from it was emitted.
…………………………………………………………………………………………………………
…………………………………………………………………………………………………………
…………………………………………………………………………………………………………
…………………………………………………………………………………………………………
Evolution from hot to cool
The discovery of the cosmic microwave background radiation is outlined on pp 87-89. This
was interpreted as the expanded (red shifted) radiation from the time around 300 000
years after the ‘big bang’ when ionised atoms recombined into atoms, and photons no
longer interacted strongly with them. This gives evidence of an evolving Universe; one that
was not the same in the past as it is now. This appeared as a bit of radio interference that
wouldn’t go away, yet it swung opinion towards an idea that several had found absurd,
even self-contradictory – that space-time had a beginning.
Page | 32
Display Material 210O
OHT 'The cosmic microwave background radiation'
The cosmic microwave background radiation
In the beginning...
...there was the Big Bang...
... the Universe is filled with
a plasma of elementary
particles, all exchanging
energy with photons of
electromagnetic radiation.
+
+
+
+
+
+
...300 000 years after the Big Bang...
Temperature: 3000 K
Typical wavelength of radiation: 1 m
As the temperature
falls, atoms form as
electrons are held in
orbit around nuclei of
protons and
neutrons.
The Universe
becomes transparent
to photons which no
longer interact so
easily with atoms and
so travel unaffected
through the Universe.
+
+
+
+
+
The decoupling of the radiation – the Universe becomes transparent to electromagnetic
radiation.
Page | 33
The cosm ic microwave background radiation
Interstellar space is
filled with a photon
‘gas’ (and some
atoms). The
temperature of this
gas is proportional to
the energy of the
photons.
The energy of a
photon is proportional
to its frequency.
Therefore the
temperature of the
photon gas is
proportional to the
frequency of the
radiation.
...13 billion years
after the Big
Bang.
Temperature: 2.7 K
Typical wavelength
of radiation: 1 mm
The Universe
expands, stretching
the wavelength of
the photons. The
greater the
wavelength, the
lower the
frequency. The
temperature of the
photon gas falls.
The expansion of the Universe stretches the wavelength of the radiation, decreasing its
frequency and therefore reducing the energy density and lowering the temperature.
Question
(a) Use the information given above to argue that the wavelength of the radiation is
inversely proportional to the temperature.
…………………………………………………………………………………………………………
…………………………………………………………………………………………………………
…………………………………………………………………………………………………………
…………………………………………………………………………………………………………
Page | 34
(b) Use the inverse relationship between wavelength and temperature, and information
supplied in Display Material 120O to show that infra-red radiation of wavelength 1 μm
formed 300 000 years after the Big Bang will now be in the microwave part of the
electromagnetic spectrum.
…………………………………………………………………………………………………………
…………………………………………………………………………………………………………
…………………………………………………………………………………………………………
…………………………………………………………………………………………………………
The expansion of the Universe
T he cosmic microwav e background radiation
no w:
cluster of
galaxies
0
R0
typical distance that increases with
the expansion of the Universe, R0
typic al wavelength, 0
then:
Rt
t
typic al wavelength, t
typical distance between clusters, R t
0 > t
R0 > Rt
so put  0 = t + 
0
R0
Rt
R0
Rt
t
=
t +  
t
=
= 1+
=

t
1+ z
This is the same effect as is seen in the r ed-shift of distant galaxies.
The expansion of spacetime and the stretching of wavelengths compared.
Page | 35
Display Material 220S
COBE'
Computer Screen 'The cosmic microwave background –
This image is a COBE image of the microwave background showing the temperature
distribution. The temperature distribution of the cosmic microwave background radiation is
over the whole celestial sphere, corrected for the effect of the Earth’s motion.
See page 89 for a colour version.
As mentioned in the introduction to this section, we have now presented the two key
pieces of evidence for the expansion and origin of the Universe: cosmological red-shift and
the cosmic microwave background.
You may have time to do Activity 140P Presentation ‘The universe’ which will involve a bit
of independent research.
The Classroom Video ‘The Big Bang’ provides a good summary for much of the work of
this chapter.
Page | 36
Questions and activities additional to those listed in the Student Notes
Section
Essential
Optional
12.1
Read A2 text pp 65-73
Qu 1-6 A2 text p 74
Question 20W Warm-up exercise 'Using
time to measure distance'
Question 30W Warm-up exercise 'Units
for distance measurement'
Question 52S Short Answer 'Doppler
detection'
Question 55S Short Answer 'Doppler
shifts in astronomy'
Question 60S Short Answer 'Binary
stars'
Question 20S Short Answer 'Measuring
distances within the solar system and beyond'
Question 40S Short Answer 'Comparing
intensities for lamps'
Question 45S Short Answer 'Jupiter and Saturn
close together in the sky'
Question 46S Short Answer 'Brighter stars
aren’t always nearer'
Question 50S Short Answer 'Trip times tell
distances'
Question 30C Comprehension 'Apparent star
brightnesses and logarithmic scales'
(hard)
Question 70D Data Handling 'Using
orbital data to calculate masses'
Question 10X Explanation–Exposition
'Logarithmic scales'
Reading 20T
Text to Read 'The ladder
of astronomical distances'
Display Material 80S Computer screen ‘Radar
images: volcano’
Display Material 100S Computer screen ‘Magic
from trip times’
File 10S
Speadsheet datatable
‘Magnitude and brightness’
File 20I
farther’
Image ‘Looking for longer, seeing
Activity 70E
Experiment 'Investigating the
measurement of distance using an ultrasonic
sensor'
Activity 80E
Experiment 'Tap-tap range
finding'
Activity 60H
Home Experiment 'Two-million
year old light: Seeing the Andromeda nebula'
The following case studies are relevant here:
‘Missing mass in the Universe’
‘Profiles from space’
These can be found in ‘Case studies: advances in
physics’ at the back of the A2 text.
12.2
Read A2 text pp 75-82
Qu 1-6 A2 text p 83
Question 100C Comprehension ‘The MichelsonMorley experiment’
Question 50C Comprehension ‘Thinking
relatively’
Question 75S Short Answer 'Relativistic
Doppler effect assuming time dilation'
Question 85S Short Answer 'Light clocks and
time dilation'
Question 150S Short Answer 'Time dilation and
length contraction for particles'
Question 50W Warm-up exercise 'When
does the speed of light matter'
Question 60W Warm-up exercise 'The
relativistic time-dilation factor γ'
Question 70S Short Answer 'Practice
with the relativistic Doppler shift equation'
Question 80S Short Answer 'Practice
with the relativistic time dilation equation'
Display material 115O OHT ‘Two-way radar
speed measurement 2’
Display material 135O OHT ‘Time dilation at v/c =
3/5’
Display material 137O OHT ‘Relativistic addition
Page | 37
Question 120S Short Answer 'Time
dilation for muons'
12.3
Read A2 text pp 84-89
Qu 1-6 A2 text p 90
Question 40W Warm-up exercise
'Cosmological expansion'
Question 95S Short Answer 'Redshifts
of quasars'
Question 100S Short Answer 'cosmic
microwave background radiation'
of velocities’
Reading 40T Text to read ‘Einstein’s 1905
relativity paper’
Reading 100T Text to read ‘Why we believe in
special relativity: experimental support for
Einstein’s theory’
Reading 100L Book list ‘Books about relativity’
Display Material 230S Computer Screen
'Looking out for longer'
Display Material 240S Computer Screen 'The
Universe at different wavelengths'
Reading 50T Text to Read 'The breakthrough
of the year'
Reading 60T Text to Read 'The sky is dark at
night: A reason to think the Universe has evolved'
Reading 70T Text to Read 'Life CV: Edwin
Powell Hubble'
Question 80D Data Handling
'Astronomical distances'
Question 90D Data Handling
'Calculating the age of the Universe'
Question 110C Comprehension
'Evidence for a hot early Universe’
Summary
Qu 1-5 A2 text p 92
These notes draw almost exclusively on the resources to be found in Advancing Physics A2 Student’s Book and CD-ROM published by
Institute of Physics Publishing in 2000 and 2008. They are intended to be used in conjunction with these resources and others not
specified.
John Mascall
The King’s School, Ely, Cambs
Page | 38