2. Neutron Stars &
Pulsars
What are they?
Neutron stars are the
remnants of collapsed
stars. They have a
mass greater than the Chandrasekhar limit [1]
which is defined to be 1.4Mʘ
(solar masses). They are formed during type II supernovae (see later), which
occur following the collapse of red supergiants.
The temperature increases as the star contracts due to gravity
causing the red supergiant to fuse progressively more massive elements, until
the predominant element in the core is iron. Iron has the greatest nuclear
stability of all the elements in the periodic table, as it has the highest binding
energy per nucleon (i.e. the energy required to overcome the strong force
holding the nucleus together, in order to separate a nucleus into its
constituent nucleons). This means that the fusion of nuclei into elements
more massive than iron would be endothermic, and hence an energetically
unstable process, as the energy released in such a reaction would be less
than the energy required fusing the nuclei. As a result of this energy
imbalance, the star can no longer produce a radiation pressure sufficient to
prevent its collapse under gravity; its collapse is instead prevented by Fermi
pressure.
This is an internal force which arises due to Heisenberg’s uncertainty
principle. The uncertainty principle states that there is an inherent
uncertainty in the product of the position and the momentum of any particle:
in other words, the greater the precision to which the position of the particle
is known, the smaller the precision its momentum can be known to, and vice
versa. The great density of the iron core means that the positions of the
electrons are defined to a high precision. This results in the electrons’
momentum increasing. According to kinetic theory, this greater momentum
results in a greater impulse imparted between colliding particles, and in a
higher frequency of collisions. The resulting force acting over an area results
in a pressure which is higher than that of an ideal gas at the same
temperature. This effect is known as the electron degeneracy (Fermi)
pressure. It is worth noting that Fermi pressure can also be explained in
terms of Pauli’s exclusion principle. This states that no two fermions can

3. (2)[4]
(1)[4]
(1)
[4]
(2)
[4]
occupy the same energy state, and it is this that prevents the iron core from
collapsing further. However a point of critical mass is reached, where the
gravitational force arising due to the mass of the iron core is greater than its
internal Fermi pressure. This causes the core to collapse.
As the star collapses in on itself, its outer layers fall in towards the
core. This results in a huge shockwave, which causes the star’s outer layers
to rebound into space. This is known as a type II supernova [2]
. A neutron-rich
remnant (i.e. the iron core) is left behind: this is a neutron star.
The star does not collapse further if it has a mass below 5.8Mʘ due to
its neutron degeneracy pressure [1]
. This is the same as the electron
degeneracy pressure, but the fermions now creating the force are neutrons.
All pulsars are neutron stars, but not all neutron stars are pulsars.
Pulsar is short for ‘pulsating radio source’. A pulsar appears to pulse due to its
rotation. At the end of a pulsar’s lifetime it will continue to rotate, although it
will no longer emit radio waves, so it is no longer a pulsar. It is due to this
rotation that we can only see certain pulsars: in order for us to observe them
their area of emission must be aligned with the Earth.
Properties of Neutron Stars
Pulsars spin very fast, due to the conservation of angular momentum
(as explained below). The fastest found neutron star has an angular velocity
of 716Hz [3]
, meaning it completes 716 complete revolutions in one second!
The conservation of angular momentum for the neutron star can be
expressed as:
nnII
where I is the object’s moment of inertia and ω is the object’s angular
velocity. The initial star had a much larger moment of inertia than the
neutron star. Thus, for angular momentum to be conserved, the neutron star
must have a larger angular velocity.
The mechanism by which neutron stars emit radio waves is called
magnetic dipole radiation. This rate of energy emission can be defined as:
2420
3
sin
43
2
m
c
P
Where c is the speed of light and θ is the angle between the magnetic dipole
m and the angular velocity ω. If the power is known, the magnetic dipole can
be used to calculate the strength of their electromagnetic fields.

4. Figure 1: Diagram of how radio waves are
radiated from a neutron star due to its strong
magnetic field.
(3)
[4]
3
0
4 R
m
B
Neutron stars have immensely large electromagnetic fields for their
size. This is because they ‘capture’ the internal magnetic field of the
collapsed star they originated from [4]
. As the size of the original star
decreases, its magnetic field lines are compacted into a smaller area,
increasing the density of field lines (otherwise known as flux), and hence
increasing the strength of the magnetic field.
The magnetic field of the
neutron star, B, approximates
to:
where μ0 is the permeability of
free space (a physical
constant), m is the magnetic
dipole of the neutron star
(which is calculated using the
neutron star’s power) and R is
its radius. Due to the alignment
of the neutron star’s magnetic
field, the direction of radio
wave emission is restricted.
This is shown in figure 1.
The electromagnetic field of the neutron star is weakest in the region
in which it emits radio waves. The direction of emission is not aligned with
the neutron star’s axis of rotation; therefore there will be periods of time
where the direction of emission is not aligned with the Earth. This is why we
receive the radio waves as pulses. This is analogous to the way a lighthouse’s
torch rotates, so the light only falls on an incoming ship at certain times, in
regular pulses, and at other times is directed elsewhere, like the coast.
The pulses are so regular and consistent that they are comparable in
their time-keeping abilities to atomic clocks [1]
. This property is extremely
useful and one of its uses, discovering gravitational waves, will be explained
later.

5. Figure 4: This is the survey chart graph that Jocelyn
Bell produced. The regular pulses can be seen on
this diagram. Each one is like a ‘tick’ of a clock,
signifying one rotation of the star.
Discovery of Neutron Stars
Neutron stars were discovered by
Jocelyn Bell (figure 2) in 1967 [1]
. Whilst
scanning the sky with a radio telescope, she
detected a regular and abnormal pulse. At first
it was believed to be interference from nearby
tractors (figure 3), or because there was no
other explanation, aliens mockingly referred to
as little green men! With only one anomaly
detected, it remained unknown for a long time
what the source was.
However with the detection of a second source, enough evidence
was provided for the existence of the theorised neutron star. The regular
pulses (figure 4) showed that the object had to be very massive but very
small at the same time. This fitted with the prediction of the highly dense
supernovae remnant, and so the existence of neutron stars was finally
confirmed.
Figure 2: Jocelyn Bell Burnell
Figure 3: A tractor (not this
one, but potentially similar)
was believed to be a
potential source of the
detected radio waves.

6. (4)[5]
Figure 8: PulsarFigure 7:
Supernova
Figure 6: Quasar/
Supermassive Black
Hole
Figure 5: The Big Bang
From left to right: Most massive events/objects known in the universe (likely to be
significant sources of gravitational waves), in decreasing order.
Background Theory of Gravitational Waves
Since neutron stars and pulsars are some of the most massive objects
in the Universe they are suitable candidates for the detection of gravitational
waves. Their enormous mass also means that they are very energetic, as
energy is proportional to mass to the power of four;
4
mE
this indicates that a small increase in mass gives rise to a much larger
increase in energy.
Binary neutron star systems might be especially useful in detecting
gravitational waves as two stars in such a system orbit each other at a
significant fraction of the speed of light. For this reason, the system is
considered to be relativistic.
Other potential sources of gravitational waves could be quasars,
supermassive black holes at the centres of galaxies, regions where the
collapse of stars is taking place and observations of the Big Bang itself.
Gravity is one of the four fundamental forces, along with the strong
force (which dominates on the scale of a nucleus), the weak force (which
governs reactions such as beta decay) and electromagnetism (responsible for
all behaviour involving charged particles). Gravitational waves are analogous
to the gravitational force as light waves are to the electromagnetic force. Just
as light exhibits wave-particle duality, where the photon is the particle
equivalent of an electromagnetic wave, gravitational waves are hypothesised
to have a particle equivalent: the graviton. Just as photons are the force

7. Figure 9: Gravitational waves emitted by
a rotating system of two massive stars.
carriers of the electromagnetic force, gravitons would be expected to
mediate gravity. If discovered, they would have to be incorporated into the
Standard Model, but this is more a problem for particle physicists than for
astrophysicists.
Difficulties in detecting gravitational waves arise since gravity is so
much weaker than the other forces: of the order 1031
times weaker than the
weak force, 1036
times weaker than electromagnetism and 1038
times weaker
than the strong force[6]
.
History of Gravitational Waves
Gravitational waves carry the information of the system that
produced them. Unlike light, they could propagate uninterrupted moments
after the Big Bang. For this reason they could allow us to observe further
back in time than has previously been possible. Gravitational waves may
provide an insight into some of the most energetic events in the cosmos, like
exploding stars and the Big Bang, and even help further our understanding of
how the universe came to be the way it looks today. They may also hold the
potential to unveil phenomena we never considered before.
Newton’s theory of gravity was based on the assumption that an
attractive force between two massive objects acts instantaneously. Two
hundred years later Einstein’s Theory of Special Relativity limited the speed
of all interactions to the speed of light (c=2.998×108
). This this is also the
maximum speed at which information can be transferred, so assuming
Einstein is correct, the force of gravity cannot be immediately felt. In these
theories gravitational waves did not exist [7]
.
In 1916, Einstein believed
that gravity was not the force
between the objects but rather a
property of space-time geometry
(space-time is the combination of
the 3-dimensional space we live in
plus time). This theory postulates
that a change in a gravitational
field will travel through the
universe at the speed of light [7]
.
Hulse and Taylor measured
the decreasing orbital period of the
binary pulsar PSR1913+16. The

8. Figure 10: The Earth orbits the Sun because of the curvature of space-time.
observation is regarded as the first indirect proof of the existence of
gravitational waves and both scientists were awarded the Nobel Prize in 1993
for this work. We have not directly detected these gravitational waves yet
but we do have strong evidence for their existence[7]
.
What are Gravitational Waves?
Gravitational waves are ripples in space-time caused by the
movement of matter. They move outward from their source, in much the
same way as ripples spreading across the surface of water.
Imagine space-time as a stretched rubber sheet [8]
. If we were to put
a heavy mass like a bowling ball on this rubber sheet, the massive object
would curve the sheet; this represents the bending of space-time caused by
gravity. If a table tennis ball was then rolled over the sheet, its path would
follow the curvature of the sheet. This example describes the way massive
objects deform space-time and affect the trajectory of masses.
The existence of graviational waves is a consequence of Albert Einstein’s
General Theory of Relativity. In his theory, the force of gravity arises due to
masses in space deforming the shape of space-time (see figure 10). The
curvature of space-time will change as massive objects move through it,
resulting in the ripples in space-time, which, according to theory, should
propagate as a wave [8]
. This is similar to the emission of electromagnetic
radiation by an accelerating electric charge.
Massive moving objects are theorised to emit waves of gravitational
radiation that carry energy away into space at the speed of light. However, by
the time these waves reach Earth they are extremely weak as their strength
decreases as they move away from the source (just as the intensity of light

9. emitted from a bulb decreases with increasing distance from the bulb. This is
one of the reasons why no direct detection of gravitational waves has yet
been made [9]
.
It is also worth noting that the faster an object moves the more
waves it produces per unit time. Additionally, more massive objects generate
more powerful waves [10]
.
Effects of a Passing Gravitational Wave
A passing gravitational wave will interact with all the particles it comes into
contact with. If we take x and y to be perpendicular axis in space-time, a
passing gravitational will cause space-time to stretch in the x direction and
compress in the y direction (or vice versa, depending on the direction of its
propagation). Any masses will mimic the deformation undergone by space-
time, as shown in figure 11.
How Do We Use Pulsars to Detect Gravitational Waves?
Indirect Measurement
Although gravitational waves have not yet directly been measured,
indirect evidence for their existence has been found. It was discovered that
the period of pulsars in a binary system decreased over time, implying a
decrease in the angular velocity of the stars, and hence a decrease in their
kinetic energy and angular momentum. The energy lost from the system was
found to equal the energy a binary pulsar system would lose when emitting
gravitational waves, as predicted by Einstein[12]
.
Figure 11: For unconnected masses initially arranged in a circle, a gravitational
wave travelling into the page stretches and compresses space-time in
perpendicular directions .As the gravitational wave hits the objects, it distorts
space-time. There are two different patterns (shown as the upper and lower
diagram) which show the effects of the two different polarisations of the waves
[11]
.

10. Figure 11: An
artist’s
impression of a
binary pulsar
system and the
gravitational
waves it induces
(which result in
energy being lost
from the system).
Use of Pulsars to Detect Gravitational Waves
This method is used to detect gravitational waves in the very low
frequency regime[14]
. The preferred pulsars used are millisecond pulsars, as
these have extremely short periods (hence more signals per minute, and
more data), which vary very little over time. This enables us to use pulsars as
precise clocks[14]
.
We can monitor the period of a large number of pulsars which are
great distances apart and occupy different regions of the sky (known as a
pulsar timing array). The mean period of a pulsar calculated over a period of
time (e.g. every month) remains almost exactly constant. It is the regularity
Figure 12: These are examples of the mean shape of pulsar signals, compared with
fluctuations in the signal due to changes in the pulsar’s interior. These profiles have
been plotted using real data collected by the Jodrell Bank Telescope.

11. of these emissions that allows pulsars to be used as clocks. If the intensity of
the signal received is plotted as a function of time, a shape results which can
be used as a control, against which later signals are compared [15]
(see figure
13).
As a gravitational wave propagates, it manifests as a ripple between
the pulsar and the detector, distorting space-time. This will alter the time the
signal takes to travel to Earth from the pulsar (see figure 14). Significant
fluctuations seen in the signals received from several of the pulsars in the
array would strongly support the existence of gravitational waves [13] [14] [16] [17]
[18] [19]
.
Using an array of pulsars is necessary in order to improve the
reliability of any findings. This is because there are several factors which
might contribute to a change in the period of a pulsar. For example: the
gradual increase of a pulsar’s period over time as the star loses kinetic energy
when it radiates, imprecisions in the clocks used on Earth and natural
fluctuations in the signal emitted, resulting from internal processes which
alter the internal structure of the star. These are examples of random error,
and so by definition they do not follow a trend. By contrast, the changes in
the pulsar signals received caused by a gravitational wave would follow a
similar pattern. Therefore, using an array allows alterations in the signal due
to gravitational waves to be differentiated from those resulting from random
Figure 14: The effects of gravitational waves on the signals of a
pulsar-timing array, with respect to Earth.

12. error [17]
. Monitoring the period of an array is also useful because the
greatest change in the received signal is seen when the gravitational wave
propagates perpendicularly to the path the light takes from the pulsar to the
Earth. This may not always be the case for one or two signals, but when
multiple pulsars are monitored the likelihood of receiving signals which have
encountered a gravitational wave edge-on increase [13]
. This is advantageous
because such signals will provide less ambiguous results.
The effect of the gravitational wave background on these pulsar
periods is most likely to be measured, although it may be possible to locate
the source of any gravitational waves which have a root mean squared
amplitude greater than that of the background. This could be due to either
the close proximity of the source, the large mass of the source or both [21]
.
Comparing the constant periods of the pulsar array to the altered time
caused by the propagation of such a gravitational wave would enable us to
calculate its polarisation and its direction of motion, which might allow us to
find the approximate position of its galaxy [22]
. However, as one arc second of
sky can contain millions and millions of galaxies (and hence ‘billions and
billions of stars’ [23]
) pinpointing the exact source of the wave would be
extremely difficult, if not impossible.
Use of Lasers to Detect Gravitational Waves
This method is used to detect gravitational waves in the high and low
frequency regimes [14]
. Two detectors are set at great distances away from
each other. The detectors are at either end of two identical perpendicular
tubes.
A laser is emitted from the furthermost end of one of the arms and is
reflected down the other perpendicular arm. The phase of the laser is such
that constructive interference occurs and the laser beam will remain exactly
the same unless a gravitational wave propagates through the system
described. Such a wave would manifest as a minute ripple in space-time,
causing one of the arms to extend and the other to contract by miniscule
amounts. This change in length should be enough to disrupt the phase of the
laser, resulting in destructive interference [20]
.
The arms must be perpendicular in order to provide the best chance
of detecting a gravitational wave, as perpendicular arms will give the
maximum difference in length, should a gravitational wave pass through.

13. The longer the arms, the greater the chances of detecting a wave, as
the extension and contraction of the arms would have a more noticeable
effect on the laser beam. Also, this would better facilitate the detection of
Gravitational waves over a greater range of frequencies. [14]
Experiments which have been/will be set up with the aim of
detecting gravitational waves include:
Pulsar Monitoring: European Pulsar Timing Array (EPTA), the North American
Nanohertz Observatory for Gravitational Waves (NANOGrav), and the Parkes
Pulsar Timing Array (PPTA) in Australia. Together, these projects form the
International Pulsar Timing Array (IPTA). Work on a new detector called the
square kilometre array (SKA) is currently in progress[15] [16]
.
Lasers: LIGO (USA), VIRGO (Italy/France), GEO (Germany/Great Britain), and
TAMA (Japan). There are plans for a group of satellites orbiting the Earth
which would reflect lasers between them (LISA), which would have a much
longer arm-length than has previously been possible (hence greater accuracy
of measurement), but no funding is available, as of yet[15] [16]
.
Implications of Gravitational Waves
Whilst the discovery of gravitational waves may (or may not) be
soon confirmed, it remains to be seen what the long term implications for
our understanding of physics will be.
Einstein’s Theory of General Relativity: If gravitational waves are detected,
the immediately obvious result would be the evidence provided in favour of
general relativity. This is already widely supported, but the detection of
gravitational waves would be yet another confirmation of the validity of
General Relativity.
One of the predictions of Relativity id that gravitational waves are
will only have two different polarisation– if more were found, a different
model (such as Einstein-anther theory[24]
) could well be true.
The consequences would be even more interesting if gravitational
waves were not found. General relativity affects much of modern physics,
and provides experimentally accurate results - if no gravitational waves were
found the theory would have to be revised.

14. Gravitons: In quantum theory, particles behave as waves and vice versa. Thus
for every type of wave (and force) there is an associated particle, such as the
photon for the electromagnetic force. The same is predicted to be true of
gravitational waves - their associated particle is known as the graviton.
From the properties of gravitational waves it should be possible to
work out some of the properties of the graviton. For example, the speed of
the gravitational wave would help us to calculate a limit on the graviton's
mass[25]
.
Inflation: Many models of the universe predict that the Big Bang was
followed by a period of rapid expansion of space-time. This is known as
inflation. As a natural consequence, this would have sent ripples throughout
space-time - gravitational waves [26] [27]
.
Not only would this provide strong evidence for gravitational waves,
but it would also allow us to study inflation itself. For example, the amplitude
of the gravitational waves produced by inflation is predicted to be
proportional to the rate of expansion of the universe.
Figure 15: This diagram shows how the universe evolved over time due to
inflation, resulting as a consequence of the big bang.

15. References
[1] An Introduction to the Sun and Stars, Green & Jones, Cambridge
[2] Physics 2 for OCR, Chadha & Sang, Cambridge
[3]www.newscientist.com/article/dn8576-fastspinning-neutron-star-smashes-speed-limit.html
[4] The Physics of Stars, A.C. Phillips, John Wiley & Sons
Pictures
[5] From private communication with Prof. Tim O’Brien (7
th
October 2014)
[6] University Physics, Young & Freedman, 13th ed, pages 1640-1641
[7] https://www.physik.hu-berlin.de/qom/research/freqref/lisa
[8]http://www.ast.cam.ac.uk/research/cosmology.and.fundamental.physics/gravitational.wav
es
[9]http://www.redorbit.com/news/space/1112850693/gravitational-wave-detector-proposed-
by-nevada-researcher-051813/
[10] http://imagine.gsfc.nasa.gov/docs/features/topics/gwaves/gwaves.html
[11] http://en.wikipedia.org/wiki/Gravitational_wave
[12] http://www.astro.cardiff.ac.uk/research/gravity/tutorial/?page=3thehulsetaylor
[13] Title: Opportunities for detecting ultra-long gravitational waves. Author: Sazhin, M. V.
Publication: Soviet Astronomy, vol. 22, Jan.-Feb. 1978, p. 36-38. Translation. Astronomicheskii
Zhurnal, vol. 55, no. 1, 1978, p. 65-68.
[14]http://www.ast.cam.ac.uk/research/cosmology.and.fundamental.physics/gravitational.wa
ves
[15] http://www.cv.nrao.edu/course/astr534/PulsarTiming.html
[16]http://en.wikipedia.org/wiki/Pulsar_timing_array
[17]Gravitational Wave Detection and Data Analysis for Pulsar Timing Arrays / by Rutger
Haasteren. http://link.springer.com/book/10.1007%2F978-3-642-39599-4
[18] http://functionspace.org/topic/298/Use-of-pulsars-for-gravitational-waves-detection-
[19]http://www.damtp.cam.ac.uk/research/gr/workshops/PGW/2009/presentations/PGW09_
Siemens.pdf
[20]http://physicsworld.com/cws/article/news/2010/jan/07/new-pulsars-could-net-
gravitational-waves
[21] Panel Reports--New Worlds, New Horizons in Astronomy and Astrophysics, by the
Committee for a Decadal Survey of Astronomy and Astrophysics Board on Physics and
Astronomy Space Studies Board Division on Engineering and Physical Sciences.
http://www.aura-
astronomy.org/news/2010/prepublication_new_worlds_new_horizons_astro2010.pdf
[22]Gravitational Wave Physics, by Kostas D. Kokkotas ,
http://www.tat.physik.unituebingen.de/~kokkotas/Teaching/NS.BH.GW_files/GW_Physics.pdf
[23] Verbal communication with Prof. Brian Cox
[24] http://relativity.livingreviews.org/Articles/lrr-2014-4/articlese7.html
[25] http://arxiv.org/abs/gr-qc/9709011
[26] http://cosmology.berkeley.edu/~yuki/CMBpol/CMBpol.htm
[27] http://arxiv.org/abs/1410.4968

16. [Figure 1] Made by Jack Hancock (based on private communication with Prof. Ben Stappers)
[Figure 2] www.physics.ox.ac.uk/astro/people/sjocelynbellburnell.htm
[Figure 3] www.telegraph.co.uk/news/uknews/law-and-order/9497022/Farm-tractor-in-low-
speed-police-chase.html
[Figure 4] https://briankoberlein.com/2014/05/14/little-green-men/
[Figure 5] http://static.ddmcdn.com/gif/big-bang-noise-670x440-130417.jpg
[Figure 6] http://www.astro.caltech.edu/~stang/m31qso
[Figure 7] http://www.riken.jp/en/research/rikenresearch/highlights/5952/
[Figure 8] http://www.universetoday.com/11671/closest-neutron-star-discovered/
[Figure 9] http://spaceplace.nasa.gov/review/lisa-g-waves/
[Figure 10] http://www.redorbit.com/news/space/1112850693/gravitational-wave-detector-
proposed-by-nevada-researcher-051813/
[Figure 11]
https://www.learner.org/courses/physics/visual/visual.html?shortname=gravitational_waves
[Figure 12] https://www.elisascience.org/articles/elisa-mission/elisa-science-goals/ultra-
compact-binaries-milky-way
[Figure 13] http://www.jb.man.ac.uk/research/pulsar/research/PulsarTiming.html
[Figure 14] http://candels-collaboration.blogspot.co.uk/2013/11/galaxy-evolution-and-
gravitation-waves.html
[Figure 15] http://bicepkeck.org/visuals.html
[Figure 16] http://www.link2portal.com/sites/default/f
iles/imagecache/lead_image_600/JeffForshaw600.jpg; and
http://manilapop.files.wordpress.com/2013/04/brian_cox_
w9vsu.jpeg
If the picture is unquoted, it is referenced here. All pictures are referred
to in chronological order.
[Picture 1] http://images.spaceref.com/news/2012/oopulsarCU0620.jpg
[Picture 2] http://www.nikhef.nl/~vdbroeck/Virgo.gif
Figure 16: A Forshaw pulsar,
emitting Cox waves (totally
fictional).