Seminar 4. Letnik

The end of the universe

Author: Grega Celcar

Menthor: Prof. dr. Tomaž Zwitter

Ljubljana, February 2013

Abstract

In this seminar I will present to you how stars evolve, which stars are important and also

which nuclear process within stars are important for distant future of the universe. Alongside

with this I will present how Interstellar Matter (ISM) will decrease and what does it mean for

distant future. In the end I will present you which stars will be lucky to live the longest.

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Kazalo

1. Stelliferious era ................................................................................................................... 1

2. Formation of stars ............................................................................................................... 2

3. Mass-luminous relation ....................................................................................................... 4

4. Nuclear reaction .................................................................................................................. 5

5. Decreasing of IMS .............................................................................................................. 7

6. Future survivors ................................................................................................................ 10

White dwarfs ......................................................................................................................... 10

Neutron stars ......................................................................................................................... 12

Brown dwarfs ....................................................................................................................... 12

Black-holes ........................................................................................................................... 13

7. Conclusion ........................................................................................................................ 13

8. Literature ........................................................................................................................... 14

1. Stelliferious era

This era started with birth of galaxies and stars and will end up with the death of galaxies and

stars. The Stelliferious era started, when universe was years old [1]. At the beginning,

first stars were born, alongside with galaxies. These first stars are referred as Population III

stars or metal free stars. These stars where made out from primordial matter which emerged

from the Big Bang as a mixture of hydrogen and helium. Astronomers assume that these stars

were very massive, due to the lack of efficient coolants of the primordial matter. Masses of

Population III stars were ranging from several hundred times of the mass of our Sun. During

their life, these stars created heavy elements all up to iron [2].

Population III stars died in around years and they dispersed their material through

universe, producing new generation of stars, called Population II and later Population I stars.

Population II or metal-poor stars includes oldest stars known, with ages in the range

years, while Population I or metal- rich stars includes young stars, some just a few

million years old, as well as some that are years old. Stars which became supernovae

also added heavier elements than iron. Population III stars were however never observed

directly, as their high mass implies that they have been extremely short-lived and none of

them could survive up to today. To see them we need to observe extremely distant galaxies

which we see as they were when the Population III stars still existed [2]. To find more direct

information about first stars, NASA will launch Webb Space Telescope, which will make

ultra-deep near-infrared surveys of the universe and of the reddened distant galaxies [3].

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2. Formation of stars

First stars or Population III stars where made from primordial matter, but stars of Population

II and stars of Population I were made from interstellar matter (ISM), which was enriched

with heavier elements by Population III. ISM surrounds enormous volume of space between

stars in the galaxy. ISM is heated and ionized by the photons emitted by all kinds of stars.

ISM is made from gas and dust and is consisting about 90 % of hydrogen and 10% of helium.

There are also small traces of heavy elements. Within ISM, we can find large molecular

clouds, which have the masses usually between solar masses, densities between

and temperature between 10 and 100 K. These clouds are called as stars

forming regions [4].

Star formation starts when the denser part of the molecular cloud core collapses under its own

gravity. The necessary minimum mass for collapse of the cloud is known as the Jeans mass

and is related to the temperature T, density and chemical composition of the cloud with the

relation (1) [4]

(1)

where μ is mean molecular weight, k is Boltzman constant and G is the gravitational constant.

For typical diffusive hydrogen cloud, we can use T=20 K and n=100

. If we assume

that cloud is entirely composed of hydrogen,

If using μ=1

and using Equation (1), the minimum mass for the cloud to collapse is 830 solar mass.

This means that, a cloud with a lower mass would not undergo a gravitational collapse.

However observations shows that stars usually tend to form in groups, ranging from binary

star systems to clusters that contain hundreds of thousands

of members. So a given ISM cloud never forms a single

star. Fragmentation of the cloud is required also by the fact

that its initial angular momentum need to be conserved

during collapse. Formation of a large number of stars from

a single cloud is able to store its initial angular

momementum into orbital momenta of the formed stars [5].

Fragmentation eventually leads to mass ovedensities which

become opaque to optical and IR radiation. These seeds of

future stars are called protostars. Opacity permits an

increase of internal temperature in protostars. In fact the

protostar heats up because its contraction releases

Figure 1: Shows how molecular

cloud collapses under gravity and

forms cloud fragments from which

protostars are made [6].

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gravitational energy and only half of the released energy is radiated away. When the central

temperature reaches approximately 15 million degrees nuclear reaction start and a true star is

born [4]. This process is shown in Figure 1.

When the gas of the protostar is still collapsing, the infalling gas releases kinetic energy in the

form of thermal energy, which increases temperature and pressure. If temperature of the core

is increased to the temperature of fusion reactions, the star is born [4].

Protostars are gravitationally bound system where the above arguments can be easily derived.

We can assume that there is a bound of spherical gas of mass M and is in hydrostatic

equilibrium as is described in Equation (2) [4]

(2)

G represent gravitational constant, r is radius from the center of the star, while is a mass

within the star at distance r from its center. Gravitational energy is described as Equation (3)

(3)

There is an introduction of spherical symmetry into Equation (3). This is because stars are

spherical symmetric systems. As we combine Equation (2) and Equation (3) and integrate

from 0 to R we get Equation (4)

. (4)

R in the Equation (4) represents radius of the star. First part on the right side of the Equation

(4) is zero, because on the radius R the pressure is zero, meanwhile when the distance from

center of the star is 0 the pressure is large. So from Equation (4) we get Equation (5).

(5)

When the star is collapsing because of the gravity, the temperature in the interior of the star

rises. This energy is called thermal energy. Thermal energy is described as Equation (6)

(6)

In Equation (6) is Boltzman constant, T is the temperature and dN represents the number

of atoms close to this temperature. dN can be described as n , while n is

a number density of the atoms and n can be described as pressure which varies with r.

Equation (7) is described like [5]

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(7)

After comparing Equation (5) and Equation (7) we get this relation between those two

energies

(8)

This relation is called virial theorem [4]. When temperature inside a star is increasing, thermal

energy increases and prevails gravitational energy. Total energy of the system can be

written as Equation (9) [4]

. (9)

Total energy is negative, which is in agreement with hypothesis that the system is bound.

When the energy is being radiated away from the stellar surface, the total energy is decreasing

according to Equation (10) [4]

. (10)

L is the luminosity of the star and is different for stars with different masses. The relation

show relation between L and M can be described with a mass-luminosity relation [4].

From Equation (10) we can calculate the time of the energy, which is radiated away from the

star. We can assume that luminosity is constant through its lifetime and for the Sun is

Time of contraction for the Sun like star is approximately years [5].

Radioactive dating established that the age of rocks on the Moon surface is over years,

which means that the Sun is older [5]. So the luminosity of a star cannot have contraction as

its energy reservoir.

3. Mass-luminous relation

We can assume that only energy transport through star is radiative and that radiation field

inside of the star is that of a black body [4]. Because of that we have net flow of photons. This

happens only when there is a pressure difference of photon densities at two different radiuses

in the star. We can write luminosity as the amount of radiation through surface, as Equation

(11.1) shows [7]

(11.1)

where is the flux of photons through a unit of area at radius r within the star. Flux of

photons can be approximated in relationship as shown in Equation (11.2)

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(11.2)

In Equation (11.2) is the energy density of photons. Because we can assume that star is a

black body celestial object, then the energy density of photons is proportional to . Further

we can approximate the mass density ρ by the total mass divided by the total volume of star.

Volume of star is proportional to . From virial theorem we can get approximation that

T

. We can now combine approximations with Equation (11.2) and put everything into

Equation (11.1), thus we get approximation mass –luminous relationship as shown (7) in

Equation (12) [7]

(12)

Equation (12) shows that the star is brighter, if stars mass is bigger and dimmer, if the stars

mass is lower. However empirical data for the stars of approximately solar chemical

composition provide for masses between 2 and 20 solar masses, for the

stars between 0.5 and 2 solar masses and for stars between 0.2-0.5 solar masses. For

the more massive stars of over 20 solar mass the relation is [4].

4. Nuclear reaction

Nuclear reaction inside of the star starts when the central temperature rises to at least K.

The first phase is called the H- burning phase and is a process of fusion of hydrogen into

helium. Stars in this phase are also on the main sequence of their evolution across the HR

diagram. This phase is also the longest evolutionary phase for the star. Nuclear reactions of

the H- burning phase can be summarized as shown in (10) [4]

where we neglected light neutrinos which take away only small amounts of energy. This

reaction originally involves four hydrogen nuclei to form one helium nucleus. Mass of helium

( is different from total mass of four hydrogen atoms (

. Because the total mass-energy relation of the system must be conserved, then

the mass loss comes out as binding energy and for H-burning phase is 26.72MeV. Binding

energy is energy which is needed to break that nucleus will break into its constituent protons

and neutrons [4].

Energy of the four hydrogen atoms combined is about 4000MeV. If we divide binding energy

and the energy of four hydrogen atoms, we get that 0.7% of the mass has changed into energy

[5].

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For simplicity we can assume that our Sun had at birth about 100% of hydrogen and that only

10% of the Sun’s mass can be converted from hydrogen to helium. Since 0.7% of the mass of

hydrogen would be converted to energy in forming helium nucleus, the amount of nuclear

energy available in the Sun would be [5]. If we use

Equation (10) and assume that luminosity of the Sun has been roughly constant, then nuclear

time for burning all hydrogen into helium is approximately [5]

When the hydrogen in the core is used up, then the nuclear reaction of H-phase ceases there.

Gravity starts to contract the helium core. Because the temperature raises high enough in the

stars inner layers the rest of hydrogen, which is still stored there, starts to fuse into helium

very rapidly. Surface layers of a such a star expand and so cool down, the star increases in

size and becomes red in color. When this will happen to our Sun it will turn into a red giant. A

few hundred million years later further contraction of the core will increase the central

temperature to very high levels ( K), so the helium’s core will be just hot enough for the

next fusion [8]. Helium will then fuse into carbon. Reaction for this fusion chain can be

summarizing as (again neglecting emerging neutrinos)

Binding energy of the carbon nucleus is 7.275 MeV and the energy of twelve hydrogen atoms

is about 11000 MeV. If we divide these two numbers, we get that 0.06% is converted from

mass to energy. Assuming that about 60% [4] of the helium is being converted into carbon

and that the luminosity at that time will be 1000 times bigger than the luminosity of the Sun is

today, the helium nuclear burning time should be approximately years. For the distant

future of the universe this simply shows that the time needed for nuclear reactions of heavier

elements is negligible, if compared to the nuclear reaction burning time of hydrogen to helium

into a star.

Since stars spend most of their lifetime mostly converting hydrogen into helium, we can

calculate the nuclear time for more and less massive stars than the Sun. For this

approximation we can use Equation (13) [5]

(13)

years is an approximate nuclear time for the Sun to lose all of its fuel. Exponent α, can

have values of 1; 2.6; 3.6; 4.5 depending in the mass of the star, as we explained above. From

the Equation (13) we can see, that the more massive the star is, the shorter is her life. Less

massive star is, thus lives longer [5].

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For example if we use a star which has about 10 solar masses, then the nuclear time for

converting hydrogen to helium is years. Nuclear time for a star with minimum mass (0.08

solar masses) to achieve fusion from hydrogen to helium is approximatelly years.

5. Decreasing of IMS

As mentioned at the beginning, one generation of stars is followed by another generation of

stars. At the beginning there were first generation stars, called Population III stars, which was

followed by Population II and later by Population I stars as shown in Figure 2. Our Sun is a

Population 1 star.

Generally, more low-mass stars than high-mass stars form from interstellar cloud fragments as

shown in Figure 3.This shows that the number of stars that forms per unit of volume is mass-

dependent and is known as the initial mass function (IMF). IMF for stars heavier than solar

mass is approximated as . Figure (3) shows that massive stars are rare

both by number and by mass, because is a declining function. The peak is somewhere in

the range between 0.1-1 solar masses [9].

Figure 3: Represents stars distribution with

different mass. The peak is somewhere between 0.1-

1 solar mass, which means that there are more low-

mass stars, than massive ones [4].

Figure 2: We can see second generation stars,

which were made from first generation stars (c).

Next the third generation of stars were born from

second generation stars (b) [8].

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IMS is turning into stars with different masses, as given by IMF. But on the other hand stars

return mass back into interstellar medium, via supernovae, stellar winds and envelope

ejections of red giants. We can calculate what fraction of mass is returned if we take one

cluster [9].

We showed above that more massive stars use up their hydrogen fuel faster than less massive

ones. So we can speak of the lowest mass of a star in a cluster which is still burning hydrogen

as a function of a cluster’s age. This is defined as turnoff mass , which is the mass of a

star, which is leaving the main sequence. As clusters ages, decreases, since lower and

lower mass stars evolve off the main sequence [9].

From IMF and turnoff mass we can figure out, what fraction of the original mass of the

cluster is within stars which are still on the main sequence (14.1) [9]

, (14.1)

where and is between 0.6-10 . Majority of stars are always on the main

sequence, even for older clusters. This is because the most common stars are those with low

masses that have not left yet the main sequence, because they have a very long lifetime. We

can calculate what fraction of the stellar mass in the cluster will remain on the main sequence

as a function of the turnoff mass. Stars that already evolved off the main sequence very

quickly burn any nuclear fuel they have and end up mostly as white dwarfs. We can assume

that all white dwarfs have a mass of 0.6 . The remaining mass already returned into

interstellar medium. From this we can then calculate what fraction of original mass is left

inside the white dwarf. This fraction can written as

, where M is in Solar masses. We can

multiply this with the middle part of the Equation (14.1) and thus we get Equation (14.2) [9]

(14.2)

This Equation (14.2) is valid only when 0.6 < <10 . is the upper mass from

which white dwarf are made. If we put into Equation (14.2) for =0.6 , we get that 14% of

the original mass is left in the form of white dwarfs.

If we now add up Equation (14.1) and Equation (14.2) we get the fraction of the original

cluster mass that remains (15) [9]

(15)

If we put the turnoff mass , then we get that 69% of the original cluster mass is

still within stars in a cluster.

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Massive stars which have mass over 10 die via supernovae explosions, which then eject

most of their mass from the cluster. Usually the cores of these stars end up as neutron stars or

black holes. Neutron stars may escape cluster, because the supernovae explosion is not always

symmetric and can kick the neutron star out from the cluster. Anyway, massive stars are quite

rare, and their remnants hold only small fraction of their initial mass. So we can assume that

all of the mass of massive stars is returned into the cluster.

We can now calculate the return fraction ζ for a stellar population back to IMS. The return

fraction is shown in Equation (16.1) [9]

(16.1)

First part of right side in Equation (16.1) represents stars from which white dwarfs are

produced. Fraction

represents the mass which is returned back to the ISM. Second

term represents stars that go supernovae and return almost all their mass back to the ISM.

Return fraction ζ can be also written as Equation (16.2) [9]

(16.2)

Turnoff mass for a very old stellar population is roughly 0.7, which gives return fraction

of 34%. Very old stellar populations return around one third of their gas back to the ISM. 20%

is returned via supernovae in a few Myrs [9]. For a younger stellar population with a larger

turn-off mass this fraction would be larger. Lifetime of a 0.7 Solar mass star can be calculated

with Equation (13) and is 35 billion years. This means that around one third of the mass is

returned back to the ISM in 35 billion years. If the mass of IMS is time dependant, we can

write a relation (17.1)

(17)

where =34 billion years. Solutions of Equation (17) are exponential and are shown in Figure

(4).

Figure 4: We can see

the decreasing of IMS

with time.

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In Figure 4 we can see that in 100 billion of years the mass of ISM reduces to 37% of the

initial mass, in 200 billion years to 7% of initial mass and in 1000 billion years to one-

millionth of initial mass. This means that this population of stars will live to years,

which is about 70 times older than is the current age of the universe.

Also if consider that in the future stars with 0.08 Solar masses will be produced, their lifetime

is also around years. After this age there will not be enough ISM for new stellar

evolution.

6. Future survivors

After there is not enough ISM for new star formation, most of the universe mass is locked

inside small stellar remnants. It is probable that 95% of the stars in the galaxy will have a

white dwarf as their remnant. Other stellar remnants are neutron stars and brown stars.

Scientist called this era Degenerate era.

White dwarfs

These stars don’t have any nuclear reaction inside of them, but are still visible for quite some

time due to the radiation from their surfaces. In white dwarfs gravity is exactly balanced by

the gradient in degeneracy pressure of electrons. This is true if the mass of white dwarf is

smaller than Chandrasekhar mass, which states that white dwarf can’t have a mass bigger than

1.4 solar masses. White dwarfs of around one solar mass have radii about 5000 km and mean

density of about

. Luminosities of observable white dwarfs are somewhere between

of Suns luminosity [4].

Cooling rate of white dwarf

When a star enters the white dwarf stage, the only significant source of energy to be radiated

is the residual ion thermal energy. Little energy can be released by further gravitational

contraction since the star has already reached a degenerate state. Thermal energy of the white

dwarf is Equation (18) [5]

(18)

Where T is the uniform interior temperature and is mass of ions. The cooling rate is given

by

. We can use Equation (10) and write L in the form of Equation (19) [4]

, (19)

where C is heat capacity. When we put everything together we obtain Equation (20.1) [4]

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. (20.1)

We integrate this Equation (20.1) from to T, where is the temperature of the core at

initial time, while T is the temperature at time t. When the white dwarf is cooled down, then

we can also assume that T, so the cooling time is [4]

. (20.2)

If we express T from Equation (20.2) and then put into Equation (19), we can get the mass-

luminous relation. This is approximated in Equation (21) [5]

(21)

Cooling time as shown on Figure 5 needed for white dwarf which has 0.001 solar luminosity

and mass around 0.6 solar mass is about years [5]. If last white dwarfs will be come to

existence when the universe will be around years old, we see they will fade away

relatively quickly after that.

While cooling, white dwarfs will face process called crystallization of the ion lattice. The

critical temperature when this can occur is called the Debye temperature ( ), below which

specific heat of the ions falls rapidly. As white dwarf cools, it crystallizes in a gradual process

that starts at the center and moves upward. The upturned “knee” in dashed line in Figure 5 at

about occurs when the cooling of nuclei begin settling into a crystalline lattice. Crystal

structure minimizes their energy as they vibrate about their average position in the lattice.

Because of this, they release their latent heat, slowing the star’s cooling and producing the

knee in the cooling curve. Later, as the temperature continues to drop, the crystalline lattice

accelerates the cooling, which leads to further energy loss of nuclei. This is reflected in the

downturn in the cooling curve. Ultimate fate for most stars is cold, dark, Earth size sphere

called as black dwarf similar to the one in Figure 6 [4].

Figure 5: Shows the cooling

time of white dwarfs. As white

dwarfs cool , aslo crytallization

starts from the core outward [4].

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Neutron stars

Neutron (Figure 7) stars have the mass around 1.5 of solar

mass. Neutron stars have the density around

and the

pressure which opposes gravity is called neutron degeneracy

pressure. These stars have the radius about 10-15 km. Because

the density is so big, it pushes protons-electron pairs together

to form a star made exclusively of neutrons [11].

The evolution of neutron stars is cooling process. At formation neutron stars have the

temperature of order K. Because of the neutrino emission and cooling, the temperature

drops to K within minutes and below K in about years [11]. Main mechanism

of cooling is the photon emission. The cooling time for neutron stars of 1.5 is shorter than

for white dwarf and is around yrs [5]. Since neutron stars are remnants of more massive

stars, their end will be long before the last stars will form.

Brown dwarfs

These stars have a mass from range of 0.02- 0.08 Solar masses. Brown dwarfs are born dead

and more resemble gigantic planets than stars. They result from gravitational collapse and

contraction of a protostar, but have insufficient mass to trigger nuclear reaction in their cores.

The only energy source for brown dwarf is gravitational contraction. Brown dwarfs are very

cool and have very low luminosity. Luminosity of a brown dwarf is close to 0.001% of Suns

luminosity. Brown dwarfs are hard to find because they are rarely found alone and are

outshone by their primary stars in a binary system [13]. In the future these so called dead

stars, will contain most of the unburned hydrogen left in the universe [1].

Figure 6: Shows a black dwarf [10] .

Figure 7: Shows a neutron star [12].

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Black-holes

If the compact stellar remnant has a mass larger than 3 then the star cannot withstand

gravitational pull and central iron core collapses. Thus the black hole is born. Density for the

black hole with a solar mass, is about

[11].

Because of a strong gravitational pull, not even light can escape the black hole. It is believed

that in every galaxy center there is a supermassive black hole, which has the mass of several

millions Solar masses. This supermassive black holes should have been born, when first

galaxies were formed. There are also some black holes, which were born out of first very

massive stars and have a mass of approximately 10 [5].

Hawking radiation

A quantum mechanical process causes black hole to evaporate. This is called the Hawking

radiation. The key for this process is the formation of virtual particles. These particles don’t

have any real life and usually they annihilate each other very quickly. But if they are made

near the black hole’s horizon one of the particles may feel gravitational pull of the black hole,

while the other particles flies away. This particle has carried away some of the black holes

mass. As the mass of the black hole is getting smaller and smaller the emission is increasing.

The end of the black hole’s evaporation proceeds very rapidly, releasing a burst of all types of

elementary particles [4]. These particles are thought to be high-energy gamma rays, together

with electrons, positrons, protons and antiprotons. The time for the black hole to evaporate is

described by Equation (21) [11]

(21)

For Solar mass black hole, Hawking evaporation is completely unimportant. When the mass

is , the timescale is shorter than the current age of the universe. This small black

hole’s could presumably have been formed during the Big Bang. Their radius is around a

fermi. For a Solar mass black hole the evaporation time reaches years [11].

7. Conclusion

There are a lot of theories what could happen after the degenerate era. Some theoretical

physicists are assuming proton decay in the future. There were some experiments to show that

proton can decay, but they were all unsuccessful. The last known experiment was taken in

Japan in 1992, called Super-Kamioka Nucleon Decay Experiment [9].

Also some science fiction fans are talking about black hole era, in which every black hole

would eventually evaporate. After the black hole era there should be dark era, where only

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particles such as, positrons, electrons, neutrinos and low-energy photons are left. But the age

of the universe in this cases should be very old and it is very is very hard to predict how our

universe will end, because there is still dark matter and especially dark energy of which we

still don’t know enough.

8. Literature

[1] F. Adams, G. Laughlin, The five ages of the universe, (The free press, 1999)

[2] Wikipedia: Metallicity. Found on 1st December 2012 on internet:

http://en.wikipedia.org/wiki/Metallicity

[3] NASA: The James Webb Space Telescope: Found on 1st of December 2012 on

internet: http://www.jwst.nasa.gov/

[4] M. Salaris, S. Cassisi, Evolution of stars and stellar populations (John Wiley & Sons,

Ltd, 2005)

[5] D. A. Ostlie, B. W. Caroll, An introduction to Modern Stellar Astrophysics, (Addison-

Wesley publishing company, Inc., 1996)

[6] Star formation: Found on 3rd

of December 2012 on internet:

http://abyss.uoregon.edu/~js/ast122/lectures/lec13.html

[7] J.F. Hewley, Foundations of Modern Cosmology, Found on 1rd

of December 2012 on

internet: http://www.astro.virginia.edu/~jh8h/Foundations/chapter5/box5b.html

[8] E. Chaisson, S. McMillan, Astronomy today, (Pearson Prentice Hall, 2005)

[9] M. Krumholz, Astronomy 112: The Physics of Stars (2009): Found on 20th

of

December 2012 on internet:

http://www.ucolick.org/~krumholz/courses/fall09_ast112/notes19.pdf

[10] Scienceblogs. Black Dwarf. Found on 15th

of December on internet:

http://scienceblogs.com/startswithabang/files/2011/02/black_dwarf.jpeg

[11] S. L. Shapiro, S. A. Teukolsky, Black hole, White Dwarfs and Neutron Stars

(Weinheim, Wiley-vch, 2004)

[12] Universe Today: Neutron star. Found on 15th

of December 2012: http://ut-

images.s3.amazonaws.com/wp-content/uploads/2008/05/foxneutronartwork.jpg

[13] S.W. Stahler, F. Palla, The formation of Stars (Weinheim,Wiley-vch, 2004)