Download Section 7 The Big Bang Theory

North Berwick High School
Department of Physics
Higher Physics
Unit 1 Our Dynamic Universe
Section 7
Big Bang Theory
Section 7
Big Bang Theory
Note Making
Make a dictionary with the meanings of any new words.
How hot are the stars?
1.
State that a star's colour depends on its surface temperature
and give a couple of examples.
2.
Briefly describe how astronomers are able to measure the
surface temperature of a star.
Cosmic microwave background
1.
State what is meant by the called cosmic microwave background
(CMB) radiation.
2.
Describe how a ‘blackbody’ spectrum of photons was produced
and state this theory was verified using the COBE satellite.
3.
Read the section on a 'fortunate accident' but do not take any
notes.
So what can the CMB tell us?
1.
What can the CMB tell us?
‘Surface of last scattering’
1.
State what is meant by a GUT force.
2.
What happened during the GUT era?
3.
What happened during the electroweak era?
4.
What happened during the particle era?
5.
Make a timeline to show these eras.
6.
State what is meant by the surface of last scattering.
The Big Bang
1.
Describe the Big Bang theory in simple terms.
Clumping
1.
Why is clumping a problem for the Big Bang theory and state
the possible solution?
Closed, open and flat
1.
State what the above terms mean and explain how they are
affected by critical density.
Many theories
1.
Read this section but do not take any notes.
How can knowing the elements in the universe
tell us about the Big Bang?
1.
Explain briefly how the Big Bang theory is supported by the
nucleosynthesis era.
2.
Why were elements heavier than helium not produced in the
early universe?
What can I look at to see the Big Bang?
1.
Read the section up to Olber's paradox. Before you read any
further, see if you can come up with an explanation.
2.
Produce a short explanation of Olbers' paradox. (The BBC
website gives a good, short explanation.)
Section 7
Big Bang Theory
Contents
Content Statements ......................................................................... 1
How hot are the stars? ..................................................................... 2
Cosmic microwave background Student activity ............................... 5
A fortunate accident: how the CMB was discovered ......................... 6
So what can the CMB tell us? ........................................................... 7
‘Surface of last scattering’................................................................ 8
Dark matter and the universe ......................................................... 10
The Big Bang .................................................................................. 10
Clumping ....................................................................................... 10
Closed, open and flat ..................................................................... 11
Many theories ................................................................................ 11
Links .............................................................................................. 12
How can knowing the elements in the universe tell us
about the Big Bang? ....................................................................... 13
Student activities ........................................................................... 14
What can I look at to see the Big Bang? .......................................... 14
Big Bang Theory Problems .............................................................. 16
Solutions ........................................................................................ 18
Content Statements
Contents
Notes
Contexts
a) The
temperature of
stellar objects.
Stellar objects emit
radiation over wide range
of wavelengths. Although
the distribution of energy
is spread over a wide range
of wavelengths, each
object emitting radiation
has a peak wavelength
which depends on its
temperature. The peak
wavelength is shorter for
hotter objects than for
cooler objects. Also,
hotter objects emit more
radiation per unit surface
area at all wavelengths
than cooler objects.
Thermal emission peaks
allow the temperature of
stellar objects to be
measured.
Remote sensing of
temperature. Investigating
the
temperature of hot
objects using infrared
sensors.
Change in colour of steel
at high temperatures.
Furnaces and kilns.
b) Evidence for the
Big Bang.
The Universe cools down as
it expands. The peak
wavelength of cosmic
microwave background
allows the present
temperature of the
Universe to be determined.
This temperature
corresponds to that
predicted after the Big
Bang, taking into account
the subsequent expansion
and cooling of the
Universe.
History of Cosmic
Microwave Background
(CMB)
discovery and
measurement.
COBE satellite.
Other evidence for the Big
Bang includes the
observed abundance of
the elements hydrogen
and helium and the
darkness of the sky
(Olber’s Paradox).
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Section 7
The Big Bang Theory
Whatʼs stellar temperature got to do with Big Bang theory?
The Big Bang theory states that the universe started with a sudden
appearance of energy at a singular point, which consequently (and very
quickly) became matter, and then expanded and cooled rapidly. The theory
therefore predicts that the universe should now, 13.7 billion years later, have
a very cool temperature. If we can measure this temperature we can see if it
accords with Big Bang theory.
If we can understand stellar temperatures, it can help us know how to find
the average temperature of the universe.
How hot are the stars?
We typically think of stars as bright white pinpoints of light in our night sky.
However, if you look carefully at the stars, even without binoculars or a
telescope, you will see a range of colours from red through yellow to blue. For
example, Betelgeuse (Orion’s armpit) looks red, Pollux (in Gemi ni) is similar to
the Sun and is yellow, and Rigel has a blue tint.
A star’s colour depends on its surface temperature. Dark red stars have
surface temperatures of about 2500 K. The surface temperature of brighter
red stars is approximately 3500 K, yellow stars, like our Sun, are roughly 5500
K, whilst blue stars range from about 10,000 to 50,000 K in surface
temperature.
Thermal emission peak
Stars emit radiation over a wide range of
wavelengths.
The graph to the right is called a thermal
emission peak which shows how the
intensity of radiation produced (y-axis) from
stars of different temperatures (the different
lines on the graph) is related to the
wavelength of light emitted from the star.
Essentially thermal emission peaks allow the
temperature of stellar objects to be
determined.
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Three details emerge from studying these peaks:
1. Stellar objects emit radiation over the complete electromagnetic spectrum.
2. Each stellar object has a peak wavelength that depends on its temperature.
3. As the temperature of the star increases:
a. There is more energy (intensity of radiation) at each wavelength
b. The peak wavelength shifts to shorter wavelengths.
Stars appear to the naked eye to be only one colour but they actually emit a
broad spectrum of colours. You can see that starlight consists of many
colours when using a prism to separate and spread the colours of the light of
the Sun, a yellow star. These colours range from red, produced by the
photons (particles of light) with the least energy, to violet, produced by the
most energetic photons.
Astronomers are able to make accurate measurements of surface
temperature by comparing the star’s apparent brightness through different
filters. The thermal radiation spectra have very distinct shapes; the
difference in apparent brightness allows astronomers to match the light
emitted to surface temperature.
Visible light is one of the bands of the electromagnetic radiation spectrum.
These range from the least energetic, radio waves, to the most energetic,
gamma rays. All six bands can be emitted by stars, but most individual stars
do not emit all of them.
Astronomers study a star’s spectrum by separating it, spre ading it out and
displaying it. The display itself is also known as a spectrum. The scientists
study thin gaps in the spectrum. When the spectrum is spread out from left
to right, the gaps appear as vertical lines. The spectra of stars have dark
absorption lines where radiation of specific energies is weak. In a few special
cases in the visible spectrum stars have bright emission lines where the
radiation of specific energies is especially strong.
An absorption line appears when a chemical element or compo und absorbs
radiation that has the amount of energy corresponding to the line. For
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example, the spectrum of the visible light coming from the Sun has a group of
absorption lines in the green part of the spectrum. Calcium in an outer layer
of the Sun absorbs light rays that would have produced the corresponding
green colours.
Although all stars have absorption lines in the visible band of the
electromagnetic spectrum, emission lines are more common in other parts of
the spectrum. For instance, nitrogen in the Sun’s atmosphere emits powerful
radiation that produces emission lines in the ultraviolet part of the spectrum.
The following webpage has an activity on this subject:
http://www.astrosociety.org/education/publications/tnl/32/starscience3.html .
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Cosmic microwave background Student activity
There is a student activity available called CMB_Kerrigan.pdf.
According to Big Bang theory, the early universe was a very small, hot and
dense place, and as it expanded, the gas within it cooled. Thus the universe
should be filled with radiation that is literally the remnant heat left over from
the Big Bang, called cosmic microwave background (CMB) radiation.
When the universe was half its current size, matter was eight times denser
and the CMB was twice as hot. So when the universe was one hundredth of
its present size, the CMB was a hundred times hotter (273 degrees above
absolute zero, 0°C). In addition to this CMB radiation, the early universe was
filled with hot hydrogen gas with a density of about 1000 atoms per cubic
centimetre. When the visible universe was only one hundred millionth its
present size, its temperature was 273 million degrees above absolute zero
and matter had a density comparable to the density of air at the Earth’s
surface. At these high temperatures, the hydrogen was completely ionised
into free protons and electrons.
Since the universe was so very hot, atoms did not form in the early universe,
only free electrons and nuclei. The CMB photons easily scattered off
electrons. So photons were sent in every direction by the early universe, just
as light scatters through a dense fog. This process of multiple scattering
produces what is called a ‘thermal’ or ‘blackbody’ spectrum of photons.
According to the Big Bang theory, the frequency spectrum of the CMB should
have this blackbody form. This was indeed measured with tremendous
accuracy by the FIRAS experiment on NASA’s COBE satellite.
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The graph below, © NASA, shows the prediction of the energy spectrum of
CMB radiation compared to the observed energy spectrum. The FIRAS
experiment measured the spectrum at 34 equally spaced points along the
blackbody curve. The error bars on the data points are so smal l that they
cannot be seen under the predicted curve in the figure! There is no
alternative theory yet proposed that predicts this energy spectrum. The
accurate measurement of its shape was another important test of the Big
Bang theory.
A fortunate accident: how the CMB was discovered
See http://www.teachersdomain.org/resource/ess05.sci.ess.eiu.microwave/ .
The existence of CMB radiation was first predicted in the1940’s, but in 1965
Arno Penzias and Robert Wilson at the Bell Telephone Laboratories in Murray
Hill, New Jersey, discovered the noise by accident. The radiation was acting
as a source of excess noise in a radio receiver they were building.
Coincidentally, researchers at nearby Princeton University, led by Robert
Dicke and including Dave Wilkinson of the WMAP science team, were devising
an experiment to find the CMB. When they heard about the Bell result they
immediately realised that the CMB had been found. Penzias and Wilson
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shared the 1978 Nobel Prize in physics for their discovery – they may not
have set out to find CMB radiation, but they were the first to do so.
Today, CMB radiation is very cold, only 2.725 K above absolute zero and this
cooling of the photon means its frequency has reduced until the radiation
shines primarily in the microwave portion of the electromagnetic spectrum
and is invisible to the naked eye. However, it can be detected everywhere we
look in the universe. In fact, if we could see microwaves, the entire sky would
glow with a brightness that was astonishingly uniform in every direction. The
temperature is uniform to better than one part in a thousand! This uniformity
is one compelling reason to interpret this radiation as re mnant heat from the
Big Bang; it would be very difficult to imagine a local source of radiation that
was this uniform. In fact, many scientists have tried to devise alternative
explanations for the source of this radiation but none have succeeded.
So what can the CMB tell us?
We have already seen that in looking at stars and galaxies we are looking at
the past. Most stars visible to the naked eye in the night sky are about 10 to
100 light years away, so we see them as they were 10 to 100 years ago. We
see Andromeda, the nearest big galaxy, as it was about 2.5 million years ago.
Observing distant galaxies with the Hubble Space Telescope, astronomers see
them as they were a few billion years after the Big Bang (between 12 and 14
billion years ago).
CMB radiation was emitted only a few hundred thousand years after the Big
Bang, and therefore long before stars or galaxies ever existed. Thus, by
studying the physical properties of the radiation, we can learn about
conditions in the universe during very early times and on very large scales,
since the radiation we see today has travelled over such a large distance.
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‘Surface of last scattering’
We’ve seen that in the early universe temperatures were so high that matter
as we typically describe it didn’t exist. The first 10 –43 s after the initial big bang
is known as the Planck era. This was a time of quantum fluctuations on a vast
energy scale. At the moment there are not any thorough explanations of this
period. Between 10 –43 s and 10 –38 s there was a period known as the GUT era.
A GUT force is a unification of the strong, weak and electromagnetic forces,
and generated a very brief but hugely significant period of expansion lasting
approximately 10 –36 s. During this time parts of the universe grew from the
size of an atom to the size of a solar system.
After this rapid expansion came the electroweak era. The universe continued
to expand, but at the same time it was cooling. The GUT force separated into
strong and electroweak forces, and gravity was also a significant force. This is
the first period after the Big Bang for which we can find evidence to support
the theoretical predictions; when CERN was able to detect for the first time
W and Z bosons, it was only able to achieve them at ener gies equivalent to a
temperature greater than 10 15 K. This is what had been predicted by Big Bang
theory for when the universe had an age of 10 –10 s.
After the electroweak era was the particle era; photons that had been the
dominant form of energy were able to form into quarks, the component parts
of protons and neutrons. By the end of this era, when the universe was
around 1 millisecond old, all the quarks had formed protons and neutrons,
and other particles such as electrons, neutrinos and possibly WIMPs had
formed.
Eventually, the universe cooled to around 3000 K – cool enough for protons
and electrons to combine to form neutral hydrogen. This occurred roughly
400,000 years after the Big Bang, when the universe was about one eleven
hundredth its present size. CMB photons are known to interact very weakly
with neutral hydrogen.
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The behaviour of CMB photons moving through the early universe is
analogous to the propagation of optical light through the Earth’s atmosphere.
Water droplets in a cloud are very effective at scattering light, while light
moves freely through clear air. Thus, on a cloudy day, we can look through
the air out towards the clouds, but cannot see through the opaque clouds.
Cosmologists studying CMB radiation can look through much of th e universe
back to when it was opaque: a view back to 400,000 years after the Big Bang.
This ‘wall of light’ is called the surface of last scattering since it was the last
time most of the CMB photons directly scattered off matter. When we make
maps of the temperature of the CMB, we are mapping this surface of last
scattering.
As shown above, one of the most striking features about the CMB is its
uniformity. Only with very sensitive instruments, such as COBE and WMAP,
can cosmologists detect fluctuations in the CMB temperature. By studying
these fluctuations, cosmologists can learn about the origin of galaxies and
their large-scale structure, and measure the basic parameters of the Big Bang
theory.
An alternative view is presented here:
http://www.deceptiveuniverse.com/Chapter5.htm.
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Dark matter and the universe
The search for dark matter is about more than explaining discrepancies in
galactic mass calculations. The missing matter problem has people
questioning the validity of current theories about how the universe formed
and how it will ultimately end.
The Big Bang
In the mid-1950s a new theory of how the universe formed emerged. The Big
Bang theory says that the universe began with a great e xplosion. The theory
evolved from Doppler shift observations of galaxies. It seems that, no matter
which direction astronomers point their telescopes, the light from the centre
of the galaxies is red-shifted. (Doppler shift caused by rotational velocity ca n
only be detected at the sides of a galaxy.) Observing red -shifted galaxies in
every direction implies expansion in all directions: an expanding universe.
The Big Bang theory is a current model for the origin of our universe and
proposes that all the matter that exists was, at one time, compressed into a
single point. The Big Bang distributed all the matter evenly in all directions.
Then the matter started to clump together, attracted by gravity, to form the
stars and galaxies that we see today. The expansion generated by the Big
Bang was great enough to overcome gravity. We still see the effects of that
force when we see red-shifted galaxies.
Clumping
One of the problems with the Big Bang theory is its failure to explain how
stars and galaxies could form in a young universe that was evenly distributed
in all directions. What started the clumping? In a smooth universe, every
particle would have the same gravitational effect on every other particle and
the universe would remain the same. But something supplied the initial
gravity to allow galaxies to form. Physicists suggest dark matter WIMPs as the
solution. Since WIMPs only affect baryon matter gravitationally, physicists
say this dark matter could be the ‘seed’ of galactic formation. ‘We don’t have
a completely successful model of galaxy formation,’ explains Walter
Stockwell, ‘but the most successful models to date seem to need plenty of
non-baryonic dark matter’.
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Closed, open and flat
There are three current scenarios that predict the future of the universe. If
the universe is closed, gravity will catch up with the expansion and the
universe will eventually be pulled back into a single point. This model
suggests an endless series of big bangs and ‘big crunches’. An open universe
has more bang than gravity – it will keep expanding forever. The flat universe
has exactly enough mass to gravitationally stop the universe from expanding,
but not enough to pull itself back in. A flat universe is said to have a critical
density of 1.
What does the expansion of the universe have to do with the missing mass?
The more mass, the more gravity. Whether the universe is closed, open or
flat depends on how much mass there is. This is where dark matter com es
into the picture. Without dark matter, critical density lies somewhere
between 0.1 and 0.01, and we live in an open universe. If there is a whole lot
of dark matter, we could live in a closed universe. Just the right amount of
dark matter and we live in a flat universe. The amount of dark matter that
exists determines the fate of the universe.
Many theories
Scientists are tossing theories back and forth. Some are skeptical of WIMPs;
particle physicists say MACHOs will never account for 90% of the univer se.
Some, like H.C. Arp, G. Burbage, F. Hoyle and J.V. Narlikan claim that
discrepancies like the dark matter problem discredits the Big Bang theory. In
the scientific journal Nature they proclaim, ‘We do not believe that it is
possible to advance science profitably when the gap between theoretical
speculation becomes too wide, as we feel it has…over the past two decades.
The time has surely come to open doors, not to seek to close them by
attaching words like ‘standard’ and ‘mature’ to theories that, judge d from
their continuing non-performance, are inadequate’. Others say there is no
missing mass. In his book What Matters: No Expanding Universe No Big Bang,
J.L. Riley claims that galactic red shift is just the effect of light turning into
matter as it ages, and not the universe expanding.
But most scientists, like Walter Stockwell, have faith in the Big Bang. ‘The
theorists will come up with all sorts of reasons why this or that can or cannot
be and change their minds every other year’, he says. ‘We experim entalists
will trudge ahead with our experiments. The Big Bang theory will outlive any
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of this stuff. It works very well as the overall framework to explain how the
universe is today’.
Now the missing mass problem is threatening humankind’s place in the
universe again. If non-baryonic dark matter does exist, then our world and
the people in it will be removed even farther from the centre. Dr Sadoulet
tells the New York Times, ‘It will be the ultimate Copernican revolution. Not
only are we not at the center of the universe as we know it, but we aren’t
even made up of the same stuff as most of the universe. We are just this
small excess, an insignificant phenomenon, and the universe is something
completely different.’
If scientists prove that non-baryonic matter does exist, it would mean that
our world and the people in it are made of something which comprises an
insignificant portion of the physical universe. A discovery of this nature,
however, probably will not affect our day-to-day process of living. ‘It’s hard
for me to imagine people getting bothered by the fact that most of the
universe is something other than baryonic. How many people even know
what baryonic means?’ comments Walter Stockwell. ‘Most of the universe is
something other than human. If their philosophy already accepts that humans
are not the center of the universe, then saying protons and neutrons aren’t
the center of the universe doesn’t seem like much of a stretch to me’.
Perhaps the only thing a dark matter discovery will give us is some
perspective.
Links
Design of the universe:
http://www.ted.com/talks/george_smoot_on_the_design_of_the_universe.html
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How can knowing the elements in the universe tell
us about the Big Bang?
The most abundant element in the universe is the simplest atom, hydrogen,
and there are many theories to support why there is so much of it. However,
just over a quarter of the ordinary matter in the universe is made up of
helium. Some of this comes from hydrogen fusion in the stars, but this source
can only account for about 10% of the observed helium in the universe. The
rest must have already been in existence in the clouds of matter that formed
the galaxies.
The helium has been created by fusion, but not in the stars. The heat for this
fusion must have come from the universe itself. Knowing the current
temperature of the CMB lets us know how hot the universe has been in the
past, and how much helium could have been made. The Big Bang theory is
again supported by this evidence. Twenty-five per cent of the universe would
have become helium during the nucleosynthesis era.
In the early parts of this 5-minute era, the universe’s temperature was 10 11 K
and protons were able to be made into slightly heavier neutrons. As
temperatures cooled, neutrons stopped being formed, but there was still
sufficient energy for protons and neutrons to fuse into deuterium (the
isotope of hydrogen containing a proton and neutron). These deuterium
nuclei fused to become helium. At about 1 minute old, the universe stopped
destroying the newly formed nuclei with gamma rays, and almost all the
neutrons had gone into creating helium. The calculated 7:1 hydrogen: helium
ratio at that time would give the elemental universe 75% hydrogen and 25%
helium at the end of the nucleosynthesis era.
So with all this energy about, why were only hydrogen and helium produced
by the Big Bang? The reason is related to how rapidly the universe was
expanding at the time. When the universe reached an age of 1 minute, the
density and temperature had fallen so low that producing heavier elements
such as oxygen or carbon was impossible. Some helium, deuterium and
protons did combine to form small amounts of lithium, but only really trace
quantities. All the heavier elements created were fused and formed in stars
nearly a billion years later.
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Student activities
Some student activities can be seen at:
http://imagine.gsfc.nasa.gov/docs/teachers/elements/imagine/activities.html .
What can I look at to see the Big Bang?
Throughout this section there have been links questioning the existence of
the Big Bang. Being the pre-eminent model of how the universe was created,
and matching with the standard model of particle physics, it is the main
target to be shot at, and unless we continue to question it then, as a theory,
it cannot grow stronger.
As much of this section of the course has been based on the knowledge
gained from using the world’s leading telescopes and detectors, there has
been little in the way of observation or practical work. However, here is a
practical we can all do to observe the universe’s origins. Go out on a clear
night, away from street lights, and ask yourself this question. Why is it dark?
Well the obvious answer is because your side of the Earth has spun away
from the Sun; it is night time. But the universe is infinite and unchanging,
with an infinite number of stars evenly arranged, radiating in all directions.
Kepler realised that if this was the case then the entire night sky would be as
bright as daytime. We know, of course, that this is not true and the scenario
is named after a German astronomer of the 1800s as Olber’s paradox.
To explain how this comes about imagine you are sitting in the middle of the
school dining hall at lunchtime. If you look in any direction you’re likely to
see another student. On a quiet day, you may be able to see through the gaps
and glimpse the walls of the room, but the busier it gets, the fewer gaps
there are. In an infinite dining hall there would be no gaps at all; in every
possible direction a student would be blocking your view.
In an infinite, uniform, unchanging universe the stars are lik e the students in
our infinite dining hall. This means we would see a star in every direction,
and everywhere in the night sky would be as bright as the surface of the Sun.
Any matter blocking our view would be heated to such a point as to glow with
the same radiance or be evaporated away.
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Our options to rationalise what we see in reality are either that there are a
finite number of stars or that the universe isn’t unchanging. Remember that
until the middle of the 20th century the historical scientific view hadn’t found
any evidence to change the literal biblical translation that creation had been
completed and that the cosmos was, therefore, unchanging. So the only
logical conclusion, as supported by Kepler, was that there were only a limited
number of stars, with nothing beyond them.
In the early 20th century this view changed. It was thought that the Milky
Way contained all the stars, but that the universe was infinite. Stars were a
little like all the tables and chairs being arranged near the middle of o ur
infinite dining hall. However, later on Hubble and others observed galaxies
distributed through space with a pretty uniform pattern and at distances
beyond the Milky Way.
Big Bang theory takes an alternative view of the universe. The theory
suggests that we can only see a limited number of stars because there was a
starting point to the universe. There may be an infinite number of stars, but
we can only see those inside our cosmological horizon. Not all the tables and
chairs are finally arranged yet.
There are other explanations for this paradox, but none other than the Big
Bang theory explains all our observations so neatly. See the following
websites:
http://www.bbc.co.uk/dna/h2g2/A765029
http://www.williams.edu/astronomy/Course-Pages/419T/olbers.pdf
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Big Bang Theory Problems
1.
The graphs below are obtained by measuring the energy emitted at
different wavelengths from an object at different temperatures.
P
Q
(a)
Which part of the x-axis, P or Q, corresponds to ultraviolet radiation?
(b)
What do the graphs show happens to the amount of energy emitted at
a certain wavelength as the temperature of the object increases?
(c)
What do the graphs show happens to the total energy radiated by the
object as its temperature increases?
(d)
Each graph shows that there is a wavelength  max at which the
maximum amount of energy is emitted.
(i)
Explain why the value of  max decreases as the
temperature of the object increases.
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(ii)
The table shows the values of  max at different temperatures of the
object.
Temperature /K
 max / m
6000
4·8 × 10 7
5000
5·8 × 10 7
4000
7 3 × 10 7
3000
9·7 × 10 7
Use this data to determine the relationship between
temperature T and  max . Choose a suitable unit for your relationship.
(e)
Use your answer to (d) (ii) to calculate:
(i)
the temperature of the star Sirius where  max is 2·7 × 10 7 m
(ii)
the value of  max for the star Alpha Crucis which has a
temperature of 23,000 K
(iii)
the temperature of the present universe when  max for
the cosmic microwave radiation is measured as 1·1 × 10 3 m.
(iv)
the approximate wavelength and type of the radiation
emitted by your skin, assumed to be at a temperature of
33 o C.
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Solutions
1.
(a)
P
(b)
Energy emitted increases
(c)
Increases
(d)
(ii)
T x  max = 2·9 × 10 3 m K
(e)
(i)
T =11, 000 K
(ii)
 max = 1·3 × 10 7 m
T = 2·6 K
 max = 9·5 × 10 6 m, infrared
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