Degeneracy

The star is prevented from further collapse by degeneracy pressure. To understand what this means we need to invoke a feature that arises from the quantum nature of matter. Particles such as electrons, neutrons and protons obey a quantum property called the Pauli Exclusion Principle. This is a quantum rule that, loosely stated, means that no two particles can be together in exactly the same quantum state at the same time. Now a star in its normal main sequence phase can be modelled more or less as an ionised ideal gas obeying an equation of state which links the gas pressure P with the number density n of the randomly moving gas particles, and the temperature T of the gas. P = nkT        (Where k is the Boltzmann's constant) However, when matter is compressed to the very high densities reached in the interior of a white dwarf, this equation of state no longer holds. The particles can no longer move randomly about and are tightly squeezed together to the extent that the electrons in neighbouring gas atoms are practically touching. For matter at high densities, the exclusion principle says that the electrons cannot be compressed any closer together and as a result they exert a powerful outward pressure that opposes any further contraction by gravity. A gas in this state is called a degenerate gas and astrophysicists say that the core of the white dwarf is held together by degenerate-electron pressure. Unlike an ideal gas, the pressure exerted by a degenerate gas does not depend on its temperature but on its density and heating it up or cooling it down makes no difference to the pressure exerted on it.

The Chandrasakar Limit

Stellar models show that the radius of a white dwarf is inversely proportional to the cube root of its mass. This means that the more massive a white dwarf is the smaller it becomes. This statement is true up to a certain limit called the Chandrasakar limit. In 1931, Subrahmanyan Chandrasakar, an Indian astrophysicist, showed that the maximum amount of mass that a white dwarf can have and still be supported by electron-degenerate pressure is about 1.4 M0. We would expect therefore, that all white dwarfs to have masses equal to or less than 1.4 M0. White dwarfs are much too faint to be seen directly with the naked eye. However, they can be observed by the effect they have on a companion star and several hundred have been detected to date.

Composition

What is the interior of a white dwarf like? As well as being very dense, as the star cools down, the material in a white dwarf forms into a crystalline lattice structure composed of ionised atoms with degenerate electrons moving freely within it. The star moves from a gaseous state through to a liquid and then becomes "solid" and in many ways ends up with properties not unlike an electrically conducting metal such as copper or silver. As millions of years pass, the white dwarf becomes colder and dimmer until its surface temperature approaches that of absolute zero ending its life as a burnt out cinder. This will be the fate of our sun, ending its life as a dark ball of oxygen and carbon, but don't worry - the sun is about halfway along the main sequence and has plenty of life in it yet!

DEGENERACY
Very simple explanation of the Pauli Exclusion Principle
http://www.chem.lgc.edu/Chm101/Aufbau/tsld006.htm

THE CHANDRASAKAR LIMIT
Explanation of the Chandrasakar Limit
http://mrcohen1.keel.physics.ship.edu/108/node5.html

Detailed biography of Subrahmanyan Chandrasakar including pictures
http://www.tamil.net/people/andrew/subra.htm

WHITE DWARFS
Simple, humorous site describing what white dwarfs are and how they behave
http://chandra.harvard.edu/xray_sources/white_dwarfs.html

                                                                                                      Crab Nebula with Neutron Star in the Centre

High mass stars (> 3 M0) are able to go through many stages of nuclear burning right through to the synthesis of iron. Iron has the highest binding energy per nucleon and no further energy can be released from the fusion of iron nuclei. Thermonuclear reactions in the core now stop and only degeneracy pressure can prevent the core from collapsing. However, for massive stars greater than 1.4 M0 but less than 3 M0 not even electron-degeneracy pressure can prevent further collapse. At some point the Chandrasakar limit is exceeded triggering a rapid collapse of the core. The central temperature rises to some 109K and the density to 1013kg m-3. Gravitational contraction is so strong that the electrons are pushed into protons forming neutrons and releasing neutrinos by the process of inverse beta decay.

p+ + e- ---> n + v

In about 1/4 of a second the density of the core becomes the same as the density of an atomic nucleus. At nuclear densities the nuclei form a superfluid composed of 80% neutrons, 10% electrons and 10% protons. Surrounding the neutron star is a crust of iron covering a solid lattice composed of neutrons and neutron-rich nuclei. The superfluid neutron core is degenerate in the same way as the electrons in a white dwarf and neutron-degeneracy pressure provides an outward force preventing any further collapse of the core enabling the neutron star to become stable. Neutron stars have high magnetic fields. The magnetic field of the original star is now concentrated over a much smaller surface area and field strengths can be as high as 108T. Since the neutron star is so dense, it has a very high surface gravity, some 1011 times that of the earth. This means that its surface is very smooth - you won't find any mountains on a neutron star! To escape from a neutron star's surface you will need to travel at about 0.8 times the speed of light.

NEUTRON STARS
Two pages of information on Neutron Stars
http://chandra.harvard.edu/xray_sources/neutron_stars.html

NASA Question and Answer site on Neutron Stars
http://imagine.gsfc.nasa.gov/docs/ask_astro/neutron_star.html

The formation of the neutron star happens extremely rapidly and once neutron-degeneracy pressure is established the core becomes rigid. The collapse of the star is dramatically halted and the infalling material bounces off the core and starts moving up towards the stars surface. A shock wave of tremendous energy is generated moving at supersonic speeds (5-10 000km s-1) blowing off the rest of the star's outer layers. The neutrinos produced by inverse beta decay swiftly travel out of the core, carrying up to 100 times more energy than is emitted as electromagnetic radiation. This gigantic explosion is called a supernova and can produce enough energy to temporarily outshine the whole galaxy! An energy of 1044 J is normally observed in supernova events.

Types of Supernovae

Astronomers have found that it is possible to categorise supernovae into two types. Type I supernovae have a sharp maximum brightness of about 1010L0 which then decreases gradually. Type II supernovae have peak luminosities on about 109 and decline much more rapidly. Type I are believed to be the result of a detonation in a close binary system involving a red giant and a white dwarf close to the Chandrasakar limit. The high surface gravity of the white dwarf attracts gas from the red giant and pushes it over the limit of 1.4 M0. The white dwarf violently collapses and its temperature increases igniting rapid carbon burning and heavier element burning, up to iron. The heat produced will eject matter from the star destroying it completely leaving no neutron star behind.

                                                                                      Before and After the 1987 Supernova

Type II are due to the formation of neutron stars and the subsequent 'core bounce' of infalling matter. In Type I the energy generated is due to thermonuclear processes whereas in Type II the energy source is gravitational. Astrophysicists are still unsure of the exact processes that give rise to supernovae and what has been described are based on computer models of stellar structure. Supernovae are not frequently seen however. An unprecedented opportunity to observe one at close range presented itself in 1987.

Supernova Remnants

In A.D. 1054 Chinese astronomers recorded the appearance of a 'guest star' that remained visible during the day for some three weeks! At night it was visible to the naked eye for a further 650 days. We now know that the Chinese astronomers saw a supernova event. This object there now is called the Crab Nebula (see first picture) and lies in the constellation of Taurus. The Crab Nebula is an example of a supernova remnant. For the closer remnants, it is possible to measure the expansion rate of the shell and therefore deduce the date of the original explosion and this is how the Crab Nebula became associated with the Chinese astronomers' records. As the expanding gas cloud slams into the interstellar medium, atoms in the gas become excited causing it to radiate electromagnetic radiation over a wide range of wavelengths from radio through to x-ray. This causes them to take on a very beautiful wispy appearance.

SUPERNOVAE
Difference between type I and II supernovae with a good selection of images
http://quark.angelo.edu/~msonntag/physics1301/supernovas.htm

List of every known supernova to occur since 1006 with information on where it occurred etc.
http://cfa-www.harvard.edu/cfa/ps/lists/Supernovae.html

Another explanation of supernovae and why they occur
http://csep1.phy.ornl.gov/guidry/violence/supernovae-info.html

Stellar explosions which occur more frequently, but are less energetic, are novae. A nova star is a star that suddenly increases in luminosity by as much as 105L0 and stays at this level for a few hours then decreases sharply and then gradually over a period of several hundred days. The energy released is not nearly so great as that from a supernova - 'only' some 1037 to 1038J! Observations of novae show that they all seem to occur in binary star systems with short orbital periods. This means that the stars are so close together that material can flow from one to the other through a region called the Roche Lobe. One model for a nova is a red giant in orbit with a white dwarf. As the red giant expands in size, matter reaches the Roche Lobe and then falls onto the white dwarf. As the matter spirals in under the white dwarf's gravity, it forms a disk of material called an accretion disk. The material that accretes onto the white dwarf's surface is mainly hydrogen which forms an envelope around the star. As more hydrogen is accreted it compresses the underlying layers heating it to temperatures exceeding 106K where explosive hydrogen burning ignites the envelope that we observe as a nova.

NOVAE
Explanation of stars turning nova
http://www.jb.man.ac.uk/~njj/glens/qj95/node5.html

During the early years of x-ray astronomy, it was discovered that some x-ray sources emitted brief but very energetic bursts lasting for several seconds before dying down. These sources are known as x-ray bursters and flare up in intervals ranging from several days to several seconds with some even shooting off bursts of energy like a machine gun! Measurements using x-ray astronomy satellites such as ROSAT have found bursters to have peak x-ray luminosities of the order of 1031W. By analysing the x-ray spectrum we can deduce that the temperature required to produce these bursts is about 3 x 107K. The accepted model of an x-ray burster involves a neutron star, and the mechanism is similar to that of a nova except that we substitute a neutron star for a white dwarf. Hydrogen from a companion falls onto a neutron star by accretion. As it does this, hydrogen becomes so hot that hydrogen burning begins before it reaches the star's surface by the CNO cycle and a helium envelope builds up where the temperatures increase to some 2 x 109K. Rapid helium burning ignites the neutron star in a helium 'flash' that releases vast amounts of x-rays in a few seconds. As more helium is accreted, the process repeats itself until all the matter from the companion star is used up. A pulsating x-ray source can be explained by infalling matter being funnelled down to the neutron star's magnetic poles where it strikes the surface with such high kinetic energy that if forms 'hot spots' which release beams of x-rays in opposite directions. If the neutron star is rotating and the Earth is in the line of sight then this is observed as a regular pulsating source.

X-RAY BURSTERS
Explanation of X-Ray Bursters with diagrams and images
http://csep1.phy.ornl.gov/guidry/violence/bursters.html

X-Ray Burster in a Globular Cluster
http://csep1.phy.ornl.gov/guidry/violence/ngc6624.html

In 1967 at the Mullard Radio Astronomy Observatory at Cambridge, a graduate student Jocelyn Bell was working with an experimental radio telescope when she noticed that the receiver was detecting regular radio pulses from a specific area in the sky. The radio pulses were extremely regular with a period of 1.33730113 s. Many more sources were soon discovered with periods ranging from 0.25 to 1.5 seconds. These objects were called Pulsars for Pulsating Radio Sources. A pulsar was discovered in the centre of the Crab Nebula, and we know that the Crab Nebula is what remains from a supernova explosion that occurred in 1054 A.D. and that neutron stars are formed when massive stars explode. Pulsars are therefore spinning neutron stars. A neutron star rotates rapidly because of the law of conservation of angular momentum. All stars rotate to a greater or lesser extent, but when suddenly reduced in size they start spinning faster. As the star rotates the beams of radiation swing round rather like a beam of light from a lighthouse. If the beam is in the direction of the earth then we see the radio component of the beam each time it points our way. It just so happens that the pulsar in the Crab is aligned at such an angle so that we get a good view. The Crab nebula pulsar is one of the fastest ever discovered with a period of 0.033 seconds and, as well as the emission at radio wavelengths, visible radiation has also been detected.

Another interesting feature of some pulsars is that they can tell us something about the structure of the neutron star. Accurate timing measurements show that the period sometimes speeds up in what astrophysicists call a 'glitch'. For this to happen means that the solid crust of the star must be brittle As the crust cools and settles, the size of the neutron star alters slightly which, because of the conservation of angular momentum, causes its rotation speed to 'flinch'. In other words the crust experiences 'starquakes' analogous to the earthquakes we experience due to movement of the earth's crust. SS 433 is a neutron star in close binary orbit with a companion and is accreting matter from it. However, the accretion rate is so high that pressure builds up along the plane of the accretion disc and is eventually relieved by the ejection of matter streaming parallel to the rotation axis of the disc.

PULSARS
Differences between Pulsars and Neutron Stars
http://hyperion.advanced.org/12713/textonly/nutrpuls.html

Questions and Answers on Pulsars
http://www.phys.vt.edu/~astrophy/faq/pulsars.html

A black hole is one of the most exotic and bizarre compact objects in modern astronomy. To understand what a black hole is we first have to look at some of the work of Albert Einstein. 1) Unlike Newtonian mechanics, space and time are not separate entities but are linked together. These four coordinates describe the space-time in which particles of matter move. 2) Space-time is curved in the presence of matter. The distribution of mass determines the degree of curvature. As a result, particles of matter move along curved paths. The geometrical path they take is what we call the 'effect of gravity'. 3) One important consequence of the curvature of space time is that the presence of mass causes light to deviate from a straight line. The more mass there is then the more the light is curved. This phenomenon is fundamental to the definition of a black hole. The effects of gravity as far as General Relativity is concerned, may be summed up by the idea that "Space-time tells matter how to move" and "matter tells space-time how to curve".

The Formation of Black Holes

For very massive stars no such mechanism exists to halt the relentless compression by gravity when the star burns out. For stars with main sequence masses greater then 10 M0 gravity squashes it to such an extent that, in theory, its density becomes infinite and its volume zero! This state of matter is called a singularity and is inaccessible to the laws of physics as we understand them. The gravitational field surrounding a black hole is so high that no radiation including light, can escape from it. As a result it appears black due to the absence of any observable radiation.

The Schwarzchild Radius

How big is a black hole? To answer this question, it is necessary to consider the amount of energy needed by mass m to escape to 'infinity' from the surface body of mass M and radius R. The result is that RSch=2GM/c2 where RSch is called the Schwarzchild radius after the German astrophysicist Karl Schwarzchild who first calculated it from the general theory of relativity. The Schwarzchild radius tells us that for a given mass, how small an object must be for it to trap light around it and therefore appear black.

The Event Horizon

The region of space where the escape velocity from the black hole is equal to the speed of light is called the event horizon and RSch is the distance from the singularity to this point. Once inside the event horizon there is no escape!

The Physical Properties of Black Holes

Once an object has passed the event horizon we cannot know anything further about its state of being. A black hole effectively removes information from the universe. However it is possible to describe every black hole by just three physical quantities: i) Its mass ii) Its electric charge iii) Its angular momentum

Inside a Black Hole

So what would conditions be like inside a black hole? We have already said that once the event horizon has been crossed there is no way of getting back. A hapless astronaut would be drawn inexorably towards the singularity. General relativity predicts that the intense gravitational field would cause time to slow down. From our spaceship we would observe our astronaut fall into the hole at a slower and slower rate until, at the event horizon, he would seem frozen in time. From our astronaut's point of view, once he passed the event horizon, gravity would soon rip him apart as he crashes towards the singularity. The laws of physics, depending as they do on a well defined framework of space and the direction of time, become meaningless and chaotic within the event horizon. However since this confusion of physical reality is unable to communicate itself across the event horizon, the laws of physics in our observable universe remain unaffected. This bizarre state of affairs has been summed up by the British mathematician Sir Roger Penrose as the law of cosmic censorship -"Thou shalt not have naked singularities". In other words, singularities do not exist without an event horizon to prevent them from interfering with the physical laws in our own universe.

How to Observe a Black Hole

As a black hole does not emit radiation of its own we can only infer its existence of one by the effect that it has on nearby objects. There are two ways in which this could happen: i)X-ray sources If a black hole forms part of a binary system then matter falling into it will become very hot as it gains kinetic energy forming an accretion disk and emitting electromagnetic radiation. As the temperature of the infalling gas reaches values in excess of 106 K then it will emit x-rays that can be detected by x-ray astronomy satellites.

ii)Gravitational Lensing

Another way a black hole might be found is through the distortion it creates in space-time. As light from a star passes close to the black hole it will be deviated either side because we know from general relativity that light is curved in the presence of a strong gravitational field. From Earth we should therefore see two images of a star, a brighter primary image on the side closest to the black hole and a fainter secondary image on the other side. (If the star, black hole and the Earth are in perfect alignment then the intensity of the images would be the same). This phenomenon is called gravitational lensing. This picture of Abell 2218 taken from the Hubble Space Telescope is an example of gravitational lensing.

Hawking Radiation

British theoretical physicist Stephen Hawking has predicted that black holes could actually evaporate! Hawking combined quantum theory with general relativity to show that near a black hole, some types of matter can split to create particle antiparticle pairs. If one of these particles then falls into the black hole while its counterpart escapes then, since the gravitational p.e. of the black hole was used to produce this pair-production, this is equivalent to the black hole losing energy since the escaping particle has carried away some of its mass. An observer would then see a steady emission of particles coming from the black hole which is known as Hawking radiation. Hawking showed that for a black hole of mass M, the rate of energy loss is proportional to M-2 so that as the black hole's mass decreases, the evaporation rate increases. As the mass dwindles to zero, the evaporation rate increases extremely rapidly, leading to a final explosive burst of elementary particles together with the emission of gamma rays. Hawking calculated that the evaporation rate tevap is approximately equal to tevap~(M/M 0)3 x1066years The estimated age of the universe is approximately 10 to 20 x 109 and if very massive black holes of mass 5 x 1011 kg were formed during the early history of the universe, then they ought to be exploding about now. The high energy gamma rays emitted should be detectable by gamma ray astronomy satellites, and gamma ray bursts have indeed been discovered. Are some of these the final moments of black holes left over from the Big Bang?

Stardeath -a summary

Main Sequence Mass/M0	End Product
0.1-0.5	White dwarf
0.5-4	Planetary Nebula, then white dwarf
4-10	Supernova, then neutron star
10-20	Supernova, then neutron star or black hole
20-60	Supernova, then black hole

BLACK HOLES
Information on Black Holes and why they should be studied
http://www.ncsa.uiuc.edu/Cyberia/NumRel/BlackHoles.html

Question and Answer section on black holes
http://skyron.harvard.edu/bh_faq.html

What would it be like to visit a black hole?
http://antwrp.gsfc.nasa.gov/htmltest/rjn_bht.html