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  

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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 ZZ 44 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:

TZ

SP 2420

3 sin43

2 mc

P »¼º

«¬ª

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.

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Figure 1: Diagram of how radio waves are radiated from a neutron star due to its strong magnetic field.

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4 RmB

SP

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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.

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.

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Figure 8: Pulsar Figure 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;

4mED 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

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

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

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].

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.

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.

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.

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.

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.

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 (7th 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.waves [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.waves [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

[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).