Pressure due to gravity is balanced by pressure of the ionized gas in the starwhich behaves like an ideal gas. Radiation leaving from the surface determinesthe luminosity of the star. A star like our sun has enough hydrogen to fuel thefusion of hydrogen to helium for some 10 billion years during which the sun shines stably.

Hydrostatic Equilibrium in an ordinary star:

How do we describe pressure exerted by a gas ?

Consider the gas in an automobile tire. You have to inflate the tire tosome pressure = 32 pounds per square inch or psi for a sedan.

The air molecules inside the tire exert this pressure. They do this bymoving around at high speeds and hitting the tire wall. The gas is inequilibrium at atmospheric temperature but high pressure.

The kinetic energy of the molecules is a measure of gas's temperature.The total number of impacts of the molecules per second on the tirewalls determines the pressure. This depends on density of moleculessqueezed into the tire – more air pushed in greater the pressure.

The relation between pressure, P, density (mass per unit volume), and temperature, T, is called the equation of state of the gas.

In a star, which is being squeezed by gravity, gas at its center is compressed, so the molecules are moving very rapidly (high temperature)and provide the pressure gradient which balances the squeezing by gravity. Equation of state depends on density and temperature.

P∝T

Equilibrium in a ' normal ' star:

The high temperature needed to balance gravity is provided by nuclear fusion energy – otherwise the star would collapse under thesqueeze of gravity.

Stars supported by Quantum pressure:

When nuclear fusion stops – the core of the star starts to collapse.As it collapses the density must rise – atoms get squeezed together.At still higher pressure Electrons in atoms are freed from atomic nuclei and form an electron gas. If the density is still higher these electrons will start getting squeezed together.

Now a new phenomena, which has no classical analogue takes over.Quantum physics requires that electrons cannot be squeezed so thattwo identical electrons occupy the same state – Pauli exclusion principle – electrons are anti-social ! They push to stay apart !

They exert a pressure, degeneracy pressure, to prevent being too close!This is enormous pressure which can support the squeezing due to gravity. But it only becomes important at very high densities.

Stars which satisfy this property, that gravity is balanced by degeneracypressure of electrons, are white dwarfs:

Degeneracy pressure is independent of temperature but depends ondensity.

Main sequence stars with masses less than 8 solar masses end up as white dwarfs after their nuclear fuel is used up. These are the cores stars of planetary nebulae.

They are small in size ( about that of the earth) but have mass close to that of the sun !

Their surface temperature is high when they are formedbut as there is no fusion to supply the energy theywill slowly cool off by radiating EM energy.

Their density is a few hundred thousand times that of the earth.

Total energy radiated or luminosity is small.

A Summary of White Dwarfs properties:

The density at which this happens is a billion kg per cubic meter !!

The surface temperature is hot: 100,000 to 200,000 degrees Kelvin

Typical mass is about 0.6 solar masses, but can extend up to 1.4 solar masswhich is the Chandrashekar limit.

Radius of the white dwarf is 6000 km – or size of the earth !!

It has no energy source.

The cores are either CO or He depending of the mass of the original star.

So it cools off slowly, unless it is a companion to a another ordinary starin a binary system. - this situation can lead to Nova or Type I SN

=109 kg /m3

Schematic of Hertzsprung-Russell or Luminosity-Temperature diagramfor stars. The population of such a diagram depends on: (1) the time that a star spends in a given location, (2) the distance to the star and obscurationby dust of visible light and (3) intrinsic luminosity of the object.

The mass of the sun and that of the white dwarfare comparable.

 Density of the sun = 1400 kg /m3

 Density of earth = 5500 kg /m3

Density of WD= 109 kg /m3

Smaller the mass of WD, largerits radius.

T= a few  times104 K

1. Gravity: Because WD have solar mass but earth radii,their gravity pull is large near its surface.

2. Accretion: If WD is a part of binary system then it canpull of mass from its companion star (accrete)and form an accretion disk.

3. X-rays: The material falling into the WD from the accretion disk get very hot from gaining gravitational energy and emits in X-rays.

Optical emission is small but it can have significant emission in X-rays as next photosshow.

Characteristics of WD

Optical image X-ray image. The brighterobject is now Sirius B whichis the WD with a very high temperature.

Luminosity of WD in optical islow because it emits mostlyin X-ray.

Accretion disk

mass transfer

The final state of Stars:

1. Ultimate fate of a star depends on it initial mass

2. White Dwarfs and Supernova Type I

3. A massive star ends with a violent explosion: called a supernova Type II

4. Matter ejected in a supernova explosion becomes a glowing supernova Remnant (SNR)

Need to know: Hydrogen is the most abundant element in our galaxy. Heliumcomes in second and Carbon and Oxygen are neck and neck for third place !

H = hydrogen atom with nucleus containing 1 protonHe = helium atom with nucleus containing 2 protons C = Carbon (nucleus has 6 protons)O = Oxygen (nucleus has 8 protons)Fe = Iron (nucleus has 26 protons)

(1) Ultimate fate of a star depends on its initial mass:

a. M < 0.4 M_sun : Too cool to fuse He to C and O. These low mass stars will eventually end as white dwarfs made of helium

b. 0.4 M_sun <M<8 M_sun : To cool to fuse C and O to heavier elements. These fairly low mass stars end as white dwarfs made up of C and O. Goes through helium-flash and asymptotic giant branch to planetary nebula to WD stages

c. M>8M_sun: Hot enough to fuse C and O to heavier elements. Very massive stars (massive enough to qualify as super-giants) are able to fuse all the way to iron, the ' end of the line ' as far as fusion is concerned.

How Nova occur in a WD which accretes material from itsorbiting normal star. When hydrogen reaches high enough temp on the surface of WD to ignite the thin hydrogen layer it make an explosion which makes a flash shining brightly, about 100,000 suns. Ejects the material from the surface creating a nova remnant. After the flash subsides it resumes accretion until the next nova.

White Dwarfs in Binary Systems : Nova and SN Type I

Novae:

As mass gets transferred on to the WD from its binary companionthe WD mass increases. If mass continues to accumulate on WDa catastrophic event can take place.

Chandrashekhar showed that if the mass of WD exceeded 1.4 M_sun, the electron-degeneracy pressure would not beable to support the pressure of gravity ! This happens when theelectrons in the WD become relativistic .

Now the WD starts to collapse and its temperature rises and getsto temperature which can fuse Carbon and create heavier elements.It ignites very rapidly throughout the star and white dwarf explodes – a white dwarf supernova.

A WD supernova – called Type I , has no H emission lines as the star which explodes is usually mostly C and sometimes mostly He.

It releases the energy equivalent of the mass of the sun !

E=0.1∗2×1030 kg×c2=1.8×1046 Joules

The factor 0.1 represents the fraction which fuses.

Nothing is left behind as a compact object!

White Dwarf Supernova:

Type I supernovae happen when a WD accretes enough mass by transfer through the Roche lobe transfer point, so as to approach the 1.4 solar mass WD limit. When WD are made mostly of carbon, its interior temperature rises high enough to ignite carbon fusion. When this happens it ignites almost instantly throughout the star and WD explodes – WD supernova. It shines brilliantly with luminosity of 10^10 suns. No hydrogen lines in their spectra.Also shown for comparison is a Massive Star Supernova, Type II

SN Type IIFATE OF A SUPER-RED GIANT STAR:

Towards the end of its life, a massive super-giant star has a central iron core, surrounded by a shell where silicon is being fused to iron, surrounded by a shell where oxygen is being fused, surrounded by a shell where carbon is being fused, surrounded by a shell where helium is being fused, surrounded by a shell where hydrogen is being fused !!

These Stars have Mass > 8 solar masses

They proceed through their stellar life much more rapidly than the Sun

They burn H using the CNO cycle as the temperatures are higher in the corethan for a less massive star, then it proceeds to red giant phase, but whentemperatures are high enough Helium burns but does not produce a He-flashrather it goes on to further synthesis in the core to C, O and heavier elements. You get the Onion structure shown on previous slide.

When synthesis of elements reaches iron, further element production is halted as iron is the most tightly bound nucleus. So if you cannot fuse iron into trans-iron nuclei (because you need energy to do that) – you do notgenerate energy to keep hydrostatic balance and synthesize elements.

An iron core is formed which gets more and more massive. When electronsin this core have sufficiently high energies they can convert protons into neutrons in nuclei, until the core is made up of entirely neutrons. Now neutron degeneracy pressure comes into play and supports gravity. No moreenergy is being produced in the core and the outside shells now collapse onto the neutron ball – all this happens in a second.

In nuclear reactions leading to fusion of iron in the inner core of this onionenormous energy is released and it escapes in the form of neutrinos out ofthe star. The iron produced cannot fuse to produce more energy and it undergoes collapse as the pressure cannot support the inward pressureof gravity. The core collapses – even quantum pressure of electrons cannot stop it.

This occurs in less than a second ! When the density of the core reaches that of nuclear density( 400 million tons per cc !!) it resists further compression and a repulsive quantum force pushes it back outwards. This shock wave heats up the outer layer.

The implosion is followed by an explosion and blows out the outer layersof the star in a shock wave – which is relativistic (5 % of the speed of light).This is what we call a supernova.

Its luminosity shoots way up and can make a previously invisible star very visible for a short time.

Supernovae are rare, luminous and relatively brief events !

Once per century somewhere in our galaxy. Luminous means with a luminosity that is ~ few×109Lsun

The Death of a super-massive star results in an explosion whichthrows debris into space (Interstellar medium -ISM) which containsall the elements which are needed for life. So the statement byCarl Sagan “ We are made up of stellar ashes “

It also emits an enormous number of high energy neutrinos in theprocess: Which were detected by groups in UCI and in Japan for theexplosion SN1987a.

It leaves behind an active nebulae emitting radiation of many differentwavelengths.

It usually leaves behind a neutron star (a star with nuclear density) whichis spinning very rapidly and acts like a light house beacon emitting periodic pulses hence the name pulsar.

It sometimes ends up as a black hole from which no radiation escapes!

The relative abundance of elements and regularity in isotopes is correctly obtained from SN II and SN Iejecta into the galaxy.

AD 1054: Chinese astronomers noted a guest star – now we know it asthe Crab nebula. From knowing the speed of the expanding shell wecan extrapolate to the time when explosion took place and it gives aboutAD 1000 !

AD 1987: SN 1987a in a nearby galaxy called SMC.

Crab Nebulacompositepicture

SN 1987a in visible light

after before

How typical SN fade away : The shapes of these curves are predicted from theory.

THE SUPERGIANT BETELGEUSE IS ONLY 160 PC. WHEN IT BECOMES A SUPERNOVA, ITS APPARENT BRIGHTNESS WILL BE (FOR A SHORTTIME) BRIGHTER THAN THE FULL MOON !!

It is also believed that SNRs are sites for the origin of Cosmic Rays, which are nuclear particles accelerated to very high energies and which influence the dynamics of our galaxy.

The acceleration of cosmic rays to high energies is done by the shock waves which last some 10,000 to 100,000 years.

We have learnt that Supernovae can be produced in more than one way !

A recapitulation:

Although SN are rare in our galaxy, they are sufficiently bright to be seen in very distant galaxies ! So we have a reasonably large sample ofobserved SN. They are classified according to their spectra.

There are two basic types:

Type I: Supernovae WITHOUT hydrogen absorption lines in their spectrum.

Type II: Supernovae WITH hydrogen absorption lines in their spectrum.

Type II supernovae are exploding massive stars whose iron cores collapse and then rebound, shock heating the outer layers of the star, which then explode outwards, just discussed.

Type I supernovae are subdivided into three subclasses Ia, Ib, and Ic !

A Type Ia supernova is caused by the transfer of matter onto a white dwarf whose mass increases while it gets denser (radius decreases)until it becomes unstable undergoes collapse and subsequentlyexplodes.

Type Ia: no H lines, no He lines, strong Silicon linesType Ib: no H lines, strong He linesType Ic: no H lines, no He lines, no Silicon lines !

Type Ib and Ic supernovae are massive stars which lost their outer layersin stellar winds before core collapse

Type Ib supernovae lost their hydrogen rich outer layer, revealing the helium rich layer immediately belowType Ic: lost their hydrogen and helium layers in mass loss and so revealingthe carbon rich layer below.

So Types Ib and Ic supernovae are essentially the same type as Type II.In all of these types the iron core of a massive star collapses and rebounds.

Neutron Stars, Pulsars and Supernova Remnants

As we have seen, Type II SN often leave behind an extremely compactobject : Neutron star

Neutron star is formed in Iron core's collapse of massive star. Its massis typically about solar mass, but its radius is only 10 km

 Volume of a neutron star is :V ns=43

×R3  ~  4×104 m3=4×1012 m3

 Thus its density is =M sun

V ns

=2×1030 Kg4×1012 m3

 or   ~ 1018 kg /m3  !!

Core of the neutron star is a liquid – surface has a crust of a lattice of iron nuclei with electrons forming a degenerate electron gas.

Neutrons provides degeneracy pressure to stabilize against gravity.

Note density of water is 1000 kg/m^3

Luminosity :Neutron star's surface temperature is about 10 million degrees KIt emits in X-rays: some

Rotation: If before collapse the star was rotating, after collapse the neutronstar will be rotating much faster to conserve angular momentum.

If the original star was a solar mass-radius object and collapsed to NSand if the original star was rotating like the sun (25 days) thenthe NS would be rotating with a period of 0.43 milli-seconds !!

Conclusion most NS should be rotating !

Spinning rate depends on initial mass, rotation and radius and final mass of the NS and its radius.

10−29 watts

Mass of Neutron Star: It generally forms after one goes beyond the stage when the star is supported by electron degeneracy pressure. This happens at the Chandrashekar limit M = 1.4 M_sun

Theory has shown that NS exits with mass which ranges from

1.2 M_sun<M < 2 or 3 times the M_sun (?) If it is more massive theneven neutron degeneracy pressure cant stop gravity and we reach a Black Hole!

NS were found accidentally by detection of regular pulsed emissionin radio waves ; Discovered by Jocelyn Bell – graduate student withHewish's special radio telescope in England. Called Pulsars.

First three pulsars had a period of about 0.253065, 1.187911 and 1.2737635 seconds – the pulsing was very precise.

Soon it was recognized ( by Gold) that they were rotating neutron stars whichemitted lighthouse beams of radio waves which were detected on earthwhen the beam swept across the earth.

Model of a pulsar:

NS is a humongous magnet withnorth and south pole. Neutrons themselves are little magnets andthey line up in the direction of the magnetic field.

The star is rotating about its spin axis and the magnetic axis is notlined up with the spin axis.

The rotating magnetic field radiates Radio waves in the directionshown by two blue cones – light house beaming. If the radio wavesintercept the earth we observe radio pulses.

2000 known pulsars

Most located in star forming regions in the disc of our galaxy

Pulsar in a Supernova Remnant - Crab

on-pulse phase off pulse phase

Hubble and Chandra images of the Crab

Hubble top sequence and Chandra X-ray bottom sequence

Binary Pulsars: Two objects in orbit around each other:

1. Two NS – one pulsar and other NS not seen as a pulsar: Taylor and Hulse Measured orbit, orbit shift, masses, and explained in terms of energy loss due to gravitational radiation

2. Pulsar and three planets circling it. Exo-solar planetary system !

3. Recently two orbiting pulsars have been observed in constellation Puppis. Periods were 23 milliseconds and 2.8 seconds !

General relativity predictions have been tested with these observations.