Universita degli Studi di Salerno

FACOLTA DI SCIENZE MATEMATICHE, FISICHE E NATURALI

Corso di Laurea in Fisica

Tesi di laurea

Compact star structure and Chandrasekhar limit

Candidato:

Fabio AratoreMatricola 0512600018

Relatore:

Ch.mo Prof. Gaetano Lambiase

Correlatore:

Dr Antonio Capolupo

Anno Accademico 2014-2015

Contents

Preface vii

1 Stellar Evolution and Classification 1

1.1 Magnitude . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.2 Color . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.3 H-R diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.4 Stellar Evolution . . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.4.1 Birth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.4.2 Mature stars . . . . . . . . . . . . . . . . . . . . . . . . . 10

1.4.3 Stellar remnants . . . . . . . . . . . . . . . . . . . . . . 11

2 The interiors of stars 15

2.1 Hydrostatic equilibrium . . . . . . . . . . . . . . . . . . . . . . 16

2.2 Mass conservation . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2.3 Equation of state . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.3.1 Classical case . . . . . . . . . . . . . . . . . . . . . . . . 20

2.3.2 Quantum statistical mechanics . . . . . . . . . . . . . . . 22

3 White dwarfs 25

3.1 Electron degeneracy . . . . . . . . . . . . . . . . . . . . . . . . 27

3.2 Electron degeneracy pressure . . . . . . . . . . . . . . . . . . . . 31

3.3 Chandrasekhar limit . . . . . . . . . . . . . . . . . . . . . . . . 33

3.3.1 The Mass-Volume relation . . . . . . . . . . . . . . . . . 33

3.3.2 Estimating the Chandrasekhar limit . . . . . . . . . . . . 35

iii

iv CONTENTS

4 Neutron stars 37

4.1 Neutron degeneracy . . . . . . . . . . . . . . . . . . . . . . . . . 37

4.2 Equation of state . . . . . . . . . . . . . . . . . . . . . . . . . . 39

4.3 Pulsar . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

4.4 Magnetars . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

5 Conclusion 49

A Nuclear reactions 51

Bibliography 55

CONTENTS v

Abstract

The objectives of this thesis are the study of the stability conditions for a star

during its main sequence phase assuming that it is in hydrostatic equilibrium.

Moreover the main features of white dwarfs and neutron stars will be studied

with particular attention to the equation of state for the degenerate matter,

the degeneracy pressure resulting from the Pauli exclusion principle (treated,

then, with the Fermi Dirac statistics). From this analysis the mass limit for

the stability of white dwarfs will be determined (the Chandrasekhar limit). Fi-

nally, the conclusive part of this thesis will be dedicated to Magnetars, special

neutron stars characterized by extremely high magnetic fields.

Gli obiettivi del presente lavoro di tesi sono lo studio delle condizioni di

stabilità di una stella durante la sua fase di sequenza principale supponendo

che essa sia in equilibrio idrostatico. Inoltre saranno studiate le caratteristiche

principali delle nane bianche e delle stelle di neutroni con particolare attenzione

all’equazione di stato per la materia degenere, alla pressione di degenerazione

derivante dal principio di esclusione di Pauli (trattata, quindi, con la statistica

di Fermi-Dirac). Da questa analisi sarà determinata la massa limite per la sta-

bilità delle nane bianche (limite di Chandrasekhar). In fine, la parte conclusiva

della tesi è dedicata alle Magnetar, particolari stelle di neutroni caratterizzate

da campi magnetici estremamente elevati.

vi CONTENTS

Preface

The etymology of the word ‘Astronomy’ implies that it was the discipline

involved in ’the arranging of the stars’. Astronomy is, at the same time, the

most ancient and the most modern science: Although we do not know who

were the first astronomers; what we do know is that the science of astronomy

was well advanced in parts of Europe by the middle of the third millennium

BC and that the Chinese people had astronomical schools as early as 2000 BC.

Today we might consider astronomy as our attempt to study and understand

celestial phenomena, part of the never-ending urge to discover order in nature.

The most important celestial object in our heavens, with no doubts, is the

Sun and is by far the most important source of energy as well as the cause of

life on Earth.

Nowadays we know that the Sun is a nearly perfect spherical ball of hot

plasma mainly made up of hydrogen and helium and much smaller quantities

of heavier elements, including oxygen, carbon, neon and iron. Our Sun, like

all the other stars, is a cosmic forge in which every elements of the periodic

table can be created.

Any star at the beginning of its life is a giant ball of hydrogen in which

center (the core) temperature reaches about 15 billions K and the hydrogen

fuses in helium. Until this thermonuclear reaction lasts the stars are in a

stable condition said main sequence. In this phase there is an hydrostatic and

energetic equilibrium that mean, in other words, the pressure of the external

layers is equal to the thermal pressure generated by nuclear reactions and the

energy produced in the core is equal to the energy radiated. The main sequence

of our Sun would last about 10 billion years: It warms and lights us since about

vii

5 billions years and it will do this still for 5 billions years. It will cease to exist

when all its fuel will end: the hydrogen. In fact the energy produced in the

thermonuclear reactions counteracts the weight of the star in order to prevent

its collapse; but when this energy will disappear it will become to shrink due

to gravity and the core starts to increase its temperature.

Stars having mass smaller that half of solar mass 1, may live for some six

to twelve trillion years, then they gradually increase in both temperature and

luminosity and accelerate the reaction rate; they will gradually turn off taking

several hundred billion more to slowly collapse leaving just its hot and dense

core said white dwarf. A white dwarf is nothing more than the ashes of the

core.

Since there is no more fuel to burn, stars cannot support themselves against

gravitational collapse by generating thermal pressure. Instead, white dwarfs

are supported by the pressure of degenerate electrons.

Bigger stars, like our Sun, ends their life in a more violent way. When there

is no hydrogen anymore the star collapse until the core reaches a temperature

necessary to fuse the helium creating carbon. The energy of this reaction push

the upper layers outwards causing expansion and creating a red giant.The

phase of red giant lasts for some hundreds of thousands years and will end

with the expulsion of the outer layers creating a big shell called planetary

nebula. During the expulsion of the external layers nitrogen and oxygen are

created. At the center of the planetary nebula it’s usually visible a white

dwarf.

The pressure generated by electrons can’t sustain mass above the Chan-

drasekhar limit of 1.4 M�. If the white dwarf having mass greater than this

limit become instable a violent contraction generates one of the most power-

ful explosion of the universe: a supernova. During the supernova explosion

the star increases its luminosity about one billion times and produce all the

elements heavier than iron. In this case the gravity is so strong that atoms

are disjointed and all the electrons fade on the core causing β decay (electron

1The solar mass is about 2× 1030 Kg and is usually indicated with the symbol 1M�

viii

capture). The collapsed object leaved by supernova explosion is a neutron

star (a rather unimaginative name describing an object made up of degenerate

neutrons).

Since the neutrons, like electrons, are fermions they obey to the Pauli

exclusion principle and generate a zero-point pressure. If the mass of the

neutron star is between 1.5 M� and 3 M� (the so called Tolman-Oppenheimer-

Volkoff limit) neither the zero-point pressure of degenerate neutrons can fight

against gravity creating a Black Hole. Black Holes are stars that could not

find any means to hold back the inward pull of gravity and therefore collapsed

to singularities; this kind of objects are so dense that their escape velocity

is greater than speed of light and nothing, neither the fastest thing in the

universe can escape from their surface.

This thesis will focus on the main features of white dwarfs and neutron

stars. First of all we will concentrate more carefully on stellar evolution and

death; then we will examine the Equation of State for the stellar matter for

both white dwarfs and neutron stars in relativistic and non-relativistic case;

finally we will talk sketchily about particular neutron stars that have an ultra

strong magnetic field of about 1011T called magnetar.

ix

x

Chapter 1

Stellar Evolution and

Classification

If seeing conditions are favorable, a view of the night sky provides a far wide

variety of celestial phenomena.

Figure 1.1: Stars pattern of

Orion in which they are clearly

visible Betelgeuse and Rigel

In addition to the Moon, some two or

three thousand tiny, twinkling points of light

(the stars) are seen, ranging in brightness

from ones easily visible just after sunset to

ones just visible when the Moon is below

the horizon and the sky background is dark-

est. Careful comparison of the bright star

with another shows that stars have different

colors; for example, in the star pattern of

Orion, one of the many constellations, Betel-

geuse is a red star in contrast to the blue

of Rigel. The apparent distribution of stars

across the vault of heaven seems random.

Stars brightness and different colors are

linked with the physical and chemical prop-

erties. We already know that at the center

of a star the temperature is so high (about

1

15× 106 K) that atoms have kinetic energy required to fuse together creating

an heavier atom whose mass is slightly lower than the sum of the two starting

atoms (see Appendix A). The lost mass ∆M became energy according to the

well known law of Einstein

E = ∆Mc2 (1.1)

where c is the velocity of light in vacuum 2.997 92× 108 m/s

1.1 Magnitude

Obviously bigger is the star, more fuel is available for nuclear reactions, more

energy is produced and hotter is the star. Now if we consider any star as

a spherical sourse radiating as a black body, its total energy output can be

determined by Stefan’s law

Etot = σT 4 (1.2)

according to it’s surface temperature and its surface area. This total output

is referred to as the stellar luminosity L and may be expressed as

L = 4πR2σT 4 (1.3)

where σ is known as Stefan’s costant and R is the radius of the star. It

appears that Stefan made a lucky guess at the law in 1879; but it was deduced

theoretically by Boltzmann in 1884. The value of σ may be evaluated by

integration of the black body curve and is given by

σ =2πk4b

15c2h3= 5.67× 10−8 W/m2K4 (1.4)

A typical value of stellar luminosity may be of the order of 1027W , that of

the Sun being 3.85× 1026 W.

The power received per unit area at the Earth depends on the stellar lumi-

2

nosity and on the inverse square of the stellar distance. If the latter is known,

the flux provided by the source may be readily calculated and expressed in

terms of W/m2. Often, the flux density from a point source such as a star is

defined as the power received per square meter per unit bandwidth within the

spectrum, with the bandpass expressed in terms of a frequency interval, with

the selected spectral interval expressed in terms of wavelength. Certainly, in

the optical region of the spectrum, it is not normal practice to measure stellar

fluxes absolutely.

The energy arriving from any astronomical body can, in principle, be mea-

sured absolutely.The brightness of any point source can be determined in terms

of the number of watts which are collected by a telescope of a given size. For

extended objects similar measurements can be made of the surface brightness.

These types of measurement can be applied to any part of the electromagnetic

spectrum.

However, in the optical part of the spectrum, absolute brightness measure-

ments are rarely made directly; they are usually obtained by comparison with

a set of stars which are chosen to act as standards. The first brightness com-

parisons were, of course, made directly by eye. In the classification introduced

by Hipparchus 1, the visible stars were divided into six groups. The brightest

stars were labeled as being of the first magnitude and the faintest which could

just be detected by eye were labeled as being of sixth magnitude. Stars with

the brightnesses between these limits were labeled as second, third, fourth or

fifth magnitude, depending on how bright the star appeared. The advent of

telescope and photometry revealed that some celestial body can be labeled

with higher magnitude than six, zero and even negative magnitude.

Since Hipparchus’ time, astronomers have extended and refined his appar-

ent magnitude scale. In the nineteenth century, it was tough that the human

eye responded to the difference in the logarithms of the brightness of two lumi-

nous objects. This theory led to a scale in which a difference of 1 magnitude

1Hipparchus of Nicaea was a Greek astronomer, geographer, and mathematician. He isconsidered the founder of trigonometry but is most famous for his incidental discovery ofprecession of the equinoxes lived between 190 and 120 B.C.

3

between two stars implies a constant ratio between their brightness. By the

modern definition, we call apparent magnitude the quantity

m = k − 2.5log10B (1.5)

where B is the star’s apparent brightness and k some constant. The value

of k is chosen conveniently by assigning a magnitude to one particular star such

as α Lyr (Vega), or set of stars, thus fixing the zero point to that magnitude

scale.

Since the energy flux that arrives on the Earth depends from the intrinsic

brightness and the distance, generally one prefers to study a quantity that is in-

dependent from the distance. Thus, using the inverse square law, astronomers

can assign an absolute magnitude M , to each star. This is defined to be the

apparent magnitude that a star would have if it were located at a distance of

10 pc 2. So if d is the distance measured in parsec, the absolute magnitude

can be expressed by the following equation

M = m+ 5 + 5log10(d) (1.6)

1.2 Color

The apparent and absolute magnitudes discussed in the Section 1.1, measured

over all wavelengths of light emitted by a star, are known as bolometric magni-

tudes and are denoted by mbol and Mbol, respectively 3. In practice, however,

detectors measure the radiant flux of a star only within certain wavelength

region defined by the sensitivity of the detector. The color of a star may be

precisely determined by using filters that transmit the stars’ light only within

a certain narrow wavelength band. In the standard UBV system, a star’s ap-

2A parsec (symbol: pc) is a unit of length used to measure the astronomically largedistances to objects outside the Solar System. One parsec is the distance at which oneastronomical unit subtends an angle of one arc-second. A parsec is equal to about 3.26light-years

3A bolometer is an instrument that measures the increase in temperature caused by aradiant flux it receives at all wavelength

4

parent magnitude is measured through three filters and is designed by three

capital letters:

• U, the star’s ultraviolet magnitude is measured through a filter centered

at 365 nm with an effective bandwidth of 68 nm

• B, the star’s blue magnitude is measured through a filter centered at

440 nm with an effective bandwidth of 68 nm

• V, the star’s visual magnitude is measured through a filter centered at

550 nm with an effective bandwidth of 89 nm

The connection between the color of light emitted by an hot object and its

temperature was first noticed in 1792 by the English maker of fine porcelain

Thomas Wedgewood. All of his ovens become red-hot at the same temperature,

independent of their size, shape and construction. Subsequent investigation by

many physicists revealed that any object with a temperature above absolute

zero emits light of all wavelengths with varying degrees of efficiency; an ideal

emitter is an object that absorbs all of the light energy incident upon it and

reradiates this energy with the characteristic spectrum show in figure. Because

an ideal emitter reflects no light, it is known as a blackbody, and the radiation

it emits is called blackbody radiation. Stars and planets are blackbodies, at

least to a rough first approximation.

Figure 1.2 shows that a blackbody of temperature T emits a continuous

spectrum with some energy at all the wavelength and that this blackbody

spectrum peaks at a wavelength λmax, which becomes shorter with increasing

temperature. The relation between λmax and T is known as Wien’s displace-

ment law:

λmaxT = 0.002 897 755 mK (1.7)

In the previous figure we can see that the blackbody radiation for 5770 K

(the sun’s surface temperature) have a maximum corresponding to λmax =

500 nm witch is in the visible light region thus we see the Sun as a yellow star.

5

Figure 1.2: Blackbody spectrum at different temperature

1.3 H-R diagram

The Hertzsprung–Russell 4 diagram, abbreviated H–R diagram or HRD, is

a scatter graph of stars showing the relationship between the stars’ absolute

magnitudes or luminosities versus their spectral classifications or effective tem-

peratures. More simply, it plots each star on a graph measuring the star’s

brightness against its temperature or color. The diagram was created in 1910

by Ejnar Hertzsprung and Henry Norris Russell and represents a major step to-

wards an understanding of stellar evolution or "the way in which stars undergo

sequences of dynamic and radical changes over time".

There are several forms of the Hertzsprung–Russell diagram, and the nomen-

clature is not very well defined. All forms share the same general layout: stars

of greater luminosity are toward the top of the diagram, and stars with higher

surface temperature are toward the left side of the diagram. The original dia-

gram displayed the spectral type of stars on the horizontal axis and the absolute

4Ejnar Hertzsprung (8 October 1873 – 21 October 1967) was a Danish chemist and as-tronomer and Henry Norris Russell (October 25, 1877 – February 18, 1957) was an Americanastronomer

6

visual magnitude on the vertical axis. The spectral type is not a numerical

quantity, but the sequence of spectral types is a monotonic series ordered by

stellar surface temperature. Modern observational versions of the chart replace

spectral type by a color index (in diagrams made in the middle of the 20th

Century, most often the B-V color) of the stars. This type of diagram is what

is often called an observational Hertzsprung–Russell diagram, or specifically a

color-magnitude diagram (CMD), and it is often used by observers. In cases

where the stars are known to be at identical distances such as with a star

cluster, the term color-magnitude diagram is often used to describe a plot of

the stars in the cluster in which the vertical axis is the apparent magnitude

of the stars: for cluster members, by assumption there is a single additive

constant difference between apparent and absolute magnitudes (the distance

modulus) for all stars. Early studies of nearby open clusters (like the Hyades

and Pleiades) by Hertzsprung and Rosenberg produced the first CMDs, ante-

dating by a few years Russell’s influential synthesis of the diagram collecting

data for all stars for which absolute magnitudes could be determined.

Most of the stars occupy the region in the diagram along the line called

the main sequence. During that stage stars are fusing hydrogen in their cores

and are in hydrostatic equilibrium. The next concentration of stars is on the

horizontal branch (helium fusion in the core and hydrogen burning in a shell

surrounding the core) characterized by luminosity much higher than the sun

and lower temperatures. At the bottom left of the HR diagram we find the

branch of white dwarfs are characterized by very high temperatures but low

surface brightness because of their size. The H-R diagram can also be used by

scientists to roughly measure how far away a star cluster is from Earth. This

can be done by comparing the apparent magnitudes of the stars in the cluster

to the absolute magnitudes of stars with known distances (or of model stars).

Contemplation of the diagram led astronomers to speculate that it might

demonstrate stellar evolution, the main suggestion being that stars collapsed

from red giants to dwarf stars, then moving down along the line of the main

sequence in the course of their lifetimes. Thus the HR diagram can be viewed

7

Figure 1.3: HR diagram

as the sets of different moments in the star’s life.

Most of the stars occupy the region in the diagram along the line called

the main sequence. During this stage stars are fusing hydrogen in their cores

and are in hydrostatic equilibrium. The next concentration of stars is on

the horizontal branch (helium fusion in the core and hydrogen burning in a

shell surrounding the core) characterized by luminosity much higher than the

sun and lower superficial temperatures that Hertzsprung called giant. This

nomenclature was natural, since the Stefan-Boltzmann law shows that

1.4 Stellar Evolution

1.4.1 Birth

Stellar evolution is the process by which a star changes during its lifetime.

Depending on the mass of the star, this lifetime ranges from a few million

8

years for the most massive to trillions of years for the least massive, which is

considerably longer than the age of the universe.

Stellar evolution starts with the gravitational collapse of a giant molecular

cloud. Typical giant molecular clouds are roughly 100 ly 5 (9.5× 1014 km)

across and contain up to 6,000,000 solar masses (1.2× 1037 kg). As it collapses,

a giant molecular cloud breaks into smaller and smaller pieces. In each of these

fragments, the collapsing gas releases gravitational potential energy as heat.

As its temperature and pressure increase, a fragment condenses into a rotating

sphere of super hot gas known as a protostar. A protostar continues to grow

by accretion of gas and dust from the molecular cloud, becoming a pre-main-

sequence star as it reaches its final mass. Further development is determined

by its mass. Protostars are encompassed in dust, and are thus more readily

visible at infrared wavelengths.

Protostars with masses less than roughly 0.08 M� (1.6× 1029 kg) never

reach temperatures high enough for nuclear fusion of hydrogen to begin. These

are known as brown dwarfs. The International Astronomical Union defines

brown dwarfs as stars massive enough to fuse deuterium at some point in

their lives (13 Jupiter masses, 2.5× 1028 kg, or 0.0125 M�). Objects smaller

than 13 Jupiter masses are classified as sub-brown dwarfs (but if they or-

bit around another stellar object they are classified as planets). Both types,

deuterium-burning and not, shine dimly and die away slowly, cooling gradually

over hundreds of millions of years. A new star will sit at a specific point on the

main sequence of the Hertzsprung–Russell diagram, with the main-sequence

spectral type depending upon the mass of the star. Small, relatively cold, low-

mass red dwarfs fuse hydrogen slowly and will remain on the main sequence

for hundreds of billions of years or longer.

5A light-year (abbreviation: ly) is a unit of length used informally to express astro-nomical distances. As defined by the IAU, the light-year is the product of the Julianyear (defined as exactly 365.25 days of 86400 SI seconds each) and the speed of light thus1 ly=9 460 730 472 580 800m

9

1.4.2 Mature stars

Recent astrophysical models suggest that red dwarfs of 0.1 M� up to 0.5 M�

may stay on the main sequence for some six to twelve trillion years, gradually

increasing in both temperature and luminosity, and take several hundred billion

more to collapse, slowly, into a white dwarf. Such stars will not become red

giants as they are fully convective and will not develop a degenerate helium

core with a shell burning hydrogen. Instead, hydrogen fusion will proceed until

almost the whole star is helium.

Stars of roughly 0.5 M� - 10 M� become red giants, which are large non-

main-sequence stars of stellar classification K or M. Red giants lie along the

right edge of the Hertzsprung–Russell diagram due to their red color and large

luminosity. Examples include Aldebaran in the constellation Taurus and Arc-

turus in the constellation of Boötes. Red giants all have inert cores with

hydrogen-burning shells: concentric layers atop the core that are still fus-

ing hydrogen into helium. Mid-sized stars are red giants during two different

phases of their post-main-sequence evolution: red-giant-branch stars, whose in-

ert cores are made of helium, and asymptotic-giant-branch stars, whose inert

cores are made of carbon. Asymptotic-giant-branch stars have helium-burning

shells inside the hydrogen-burning shells, whereas red-giant-branch stars have

hydrogen-burning shells only. In either case, the accelerated fusion in the

hydrogen-containing layer immediately over the core causes the star to expand.

This lifts the outer layers away from the core, reducing the gravitational pull

on them, and they expand faster than the energy production increases. This

causes the outer layers of the star to cool, which causes the star to become

redder than it was on the main sequence.

During their helium-burning phase, very high-mass stars with more than

9 M� expand to form red supergiants. Once this fuel is exhausted at the core,

they continue to fuse elements heavier than helium. The core contracts until

the temperature and pressure suffice to fuse carbon. This process continues,

with the successive stages being fueled by neon, oxygen, and silicon. Near the

end of the star’s life, fusion continues along a series of onion-layer shells within

10

the star. Each shell fuses a different element, with the outermost shell fusing

hydrogen; the next shell fusing helium, and so forth.The final stage occurs

when a massive star begins producing iron. Since iron nuclei are more tightly

bound than any heavier nuclei, any fusion beyond iron does not produce a

net release of energy the process would, on the contrary, consume energy. In

relatively old, very massive stars, a large core of inert iron will accumulate in

the center of the star causing Supernovae explosion.

1.4.3 Stellar remnants

After a star has burned out its fuel supply, its remnants can take one of three

forms, depending on the mass during its lifetime:

• When the red giant phase ends, the outer layers are expelled, leaving vis-

ible the hot core: the white dwarf. For a star of 1 M�, the resulting white

dwarf is of about 0.6 M�, compressed into approximately the volume of

the Earth. White dwarfs are stable because the inward pull of gravity

is balanced by the degeneracy pressure of the star’s electrons, a conse-

quence of the Pauli exclusion principle. Electron degeneracy pressure

provides a rather soft limit against further compression (Chandrasekhar

limit). With no fuel left to burn, the star radiates its remaining heat into

space for billions of years. A white dwarf is very hot when it first forms,

more than 100,000 K at the surface and even hotter in its interior. It is so

hot that a lot of its energy is lost in the form of neutrinos for the first 10

million years of its existence, but will have lost most of its energy after a

billion years. In the end, all that remains is a cold dark mass sometimes

called a black dwarf. However, the universe is not old enough for any

black dwarfs to exist yet. If the white dwarf’s mass increases above the

Chandrasekhar limit, which is 1.4 M� then electron degeneracy pressure

fails due to electron capture and the star collapses.

• A super red giant core has greater than Chandrasekhar limit thus the

gravity is stronger than the degenerate electron pressure so the last ones

11

Figure 1.4: Schematization of all possible stages of a star’s life from birth todeath

fade onto nucleus converting the great majority of the protons into neu-

trons. Now, the nucleus has a radius of about 10−15m while the entire

atom have radius of about 10−10m: almost all of the atom is empty space.

The electromagnetic forces keeping separate nuclei apart are gone, and

most of the core of the star becomes a dense ball of contiguous neutrons.

The neutrons, being themselves fermions, resist further compression by

the Pauli Exclusion Principle, in a way analogous to electron degener-

acy pressure, but stronger. These stars, known as neutron stars, are

extremely small (on the order of radius 10 km, no bigger than the size

of a large city) and are phenomenally dense (about 1018Kg/m3 that is 1

thousand billion tons). Their period of rotation shortens dramatically as

the stars shrink (due to conservation of angular momentum); observed

rotational periods of neutron stars range from about 1.5 milliseconds

(over 600 revolutions per second) to several seconds. When these rapidly

rotating stars’ magnetic poles are aligned with the Earth, we detect a

pulse of radiation each revolution. Such neutron stars are called pulsars,

and were the first neutron stars to be discovered.

• If the mass of the stellar remnant is high enough, the neutron degeneracy

pressure will be insufficient to prevent collapse below the Schwarzschild

12

radius. The stellar remnant thus becomes a black hole. The mass at

which this occurs is not known with certainty, but is currently estimated

at between 2 M� and 3 M� called Tolman-Oppenheimer-Volkoff limit.

Black holes are predicted by the theory of general relativity. According

to classical general relativity, no matter or information can flow from the

interior of a black hole to an outside observer, although quantum effects

may allow deviations from this strict rule. The existence of black holes

in the universe is well supported, both theoretically and by astronomical

observation.

13

14

Chapter 2

The interiors of stars

Analysis of that light, collected by ground-based and space-based telescopes,

enables astronomers to determine a variety of quantities related to the outer

layers of stars, such as effective temperature, luminosity, and composition.

However, with the exceptions of the ongoing detection of neutrinos from the

Sun or the one-time detection from Supernova 1987A, no direct way exist to

observe the central regions of stars.

To deduce the detailed internal structure of stars requires the generation of

computer models that are consistent with all known physical laws and that ul-

timately agree structure was understood by the first half of the 20th century, it

wasn’t until the 1960s that sufficiently fast computing machines became avail-

able to carry out all of the necessary calculations. Arguably one of the greatest

successes of theoretical astrophysics has been the detailed computer modeling

of stellar structure and evolution. However, despite all of the successes of such

calculations, numerous questions remain unanswered. The solution to many of

these problems requires a more detailed theoretical understanding of physical

processes in operation in the interiors of stars, combined with even greater

computational power.

The theoretical study of stellar structure, coupled with observational data,

clearly shows that stars are dynamic objects, usually changing at an imper-

ceptibly slow rate by human standards, although they can sometimes change

in very rapid and dramatic ways, such as during a supernova explosion. That

15

such changes must occur can be seen by simply considering the observed en-

ergy output of a star. In the Sun 3.839× 1026 J of energy is emitted every

second. This rare of energy output would be sufficient to melt a 0℃ block of

ice measuring 1 AU x 1 Km x 1 Km in only 0.2 s, assuming that the absorption

of the energy was 100% efficient. Because stars do not have infinite supplies

of energy, they must eventually use up their reserves and die. tellar evolution

is the result of a constant fight against the relentless pull of gravity.

2.1 Hydrostatic equilibrium

The gravitational force is always attractive, implying that an opposing force

must exist if a star is to avoid collapse: this force is pressure. To calculate

how the pressure must vary with depth, consider a cylinder of mass dm whose

base is located at distance r from the center of a spherical star (see Fig. 2.1).

The areas of the top and bottom of the cylinder are each A and the cylinder’s

height is dr. Furthermore, assume that the only forces acting on the cylinder

are gravity and the pressure force, which is always normal to the surface and

may vary with distance from the center of the star. Using Newton’s second

law we have:

dmd2r

dt2= Fg + FP,t + FP,b (2.1)

where Fg < 0 is the gravitational force directed towards the center and

FP,t and FP,b are the pressure forces on the top and bottom of the cylinder

respectively; moreover the pressure on the top base is directed like the gravity

(so FP,t < 0) and FP,b > 0 because is directed toward the surface. We can

write FP,b = −(FP,b + dFP ) where dFP is the correction that accounts for the

change in force due to a change in r.

As we know, the gravitational force on a small mass dm located at a dis-

tance r from the center of a spherically symmetric mass is

Fg = −GMrdm

r2(2.2)

16

Figure 2.1: In a static star the gravitational force on a mass element is exactlycanceled by the outward force due to a pressure gradient in the star. A cylinderof mass dm is located at distance r from the center of the star. The heightof the cylinder is dr and the areas on the top and bottom are both A. Thedensity of the gas in assumed to be ρ at that position.

17

where Mr is the mass inside the sphere of radius r, often referred to as the

interior mass (remembering Gauss’Theorem). Taking into account even the

definition of pressure dFP = AdP and the definition of mass density dm =

ρdV = ρAdr, substituting all these results in Eq. (2.1)

ρAdrd2r

dt2= −GMrρAdr

r2− AdP (2.3)

Finally, dividing through by the volume of the cylinder, we have

ρd2r

dt2= −GMrρ

r2− dP

dr(2.4)

This is the equation for the radial motion of the cylinder, assuming spherical

symmetry. If we assume further that the star is static, then the acceleration

must be zero, so Eq. 2.4 reduces to

dP

dr= −GMrρ

r2(2.5)

This equation represent the condition of hydrostatic equilibrium and is one

of the fundamental equations of stellar structure for spherically symmetric

object under the assumption that accelerations are negligible. Note that Eq.

(2.5) clearly indicate that in order for a star to be static, a pressure gradient

must exist to counteract the force of gravity. It is not the pressure that supports

a star, but the change in pressure with radius. Furthermore, the pressure must

decrease with increasing radius; the pressure is necessarily larger in the interior

than it is near surface.

2.2 Mass conservation

A second relationship involving mass, radius, and density also exist. Again,

for a spherically symmetric star, consider a shell of mass dMr and thickness

dr, located a distance r from the center. Assuming that the shell is sufficiently

this, the volume of the shell is approximately dV = 4πr2dr. If the local density

of the gas is ρ, the shell’s mass is given by

18

Figure 2.2: A spherically symmetric shell of mass dM , having a thickness drand located at distance r from the center of the star. The local density of theshell is ρ.

dMr = 4πρr2dr (2.6)

Rewriting, we arrive at the mass conservation equation

dMr

dr= 4πρr2 (2.7)

which dictates how the interior mass of a star must change with distance

from the center. Equation (2.7) in the second of the fundamental equations of

stellar structure.

2.3 Equation of state

Up to this point no information has been provided about the origin of the pres-

sure term required by Eq. (2.5). To describe this macroscopic manifestation

of particle interactions, it is necessary to derive a pressure equation of state of

the material. Such an equation of state relates the dependence of pressure on

other fundamental parameters of the material. One well-known example of a

pressure equation of state is the ideal gas law, often expressed as PV = NkbT

where kb = 1.380 648 8× 10−23 J/K is Boltzmann’s constant.

19

The physical state of Stellar interior matter is given by the fundamental

hypothesis that the matter is in local thermodynamical equilibrium, this means

that it is locally described by the usual thermodynamical laws, the kinetic

theory and the statistical mechanics.

From the kinetic theory, we know that in a gas with np particles per unit

volume having momenta between p and p + dp, the pressure exerted on the

wall of the container will be

P =1

3

∫ +∞

0

nppvdp (2.8)

This expression, which is sometimes called the pressure integral, makes it

possible to compute the pressure, given some distribution function, npdp.

2.3.1 Classical case

Equation (2.8) is valid for both massive and massless particles (such as pho-

tons) traveling at any speed. For the special case of massive, non-relativistic

particles, we may use p = mv to write the pressure integral as

P =1

3

∫ +∞

0

mnvv2dv (2.9)

where nvdv is the number of particles per unit volume having speeds be-

tween v and v + dv. This function is dependent on the physical nature of

the system being described. In the case of an ideal gas, nvdv is the Maxwell-

Boltzmann velocity

nvdv = n(m

2πkbT)32 e− mv2

2kbT 4πv2dv (2.10)

where n =∫ +∞0

nvdv is the particle number density. Substituting into the

pressure integral and remembering that n = NV

it yelds to the ideal gas law

PV = NkbT (2.11)

In astrophysical applications it is often convenient to express the ideal gas

20

law in an alternative form. Since n is the particle number density, it is clear

that it must be related to the mass density of the gas. Allowing for a variety

of particles of different masses, it is then possible to express n = ρm

where m is

the average mass of a gas particle; moreover if we define the mean molecular

weight as µ = mmH

, the ideal gas law becomes

Pg =ρkbT

µmH

(2.12)

where mH = 1.673 532 499× 10−27 Kg is the mass of the hydrogen atom.

The mean molecular weight is just the average mass of a free particle in the

gas, in units of the mass of hydrogen. The mean molecular weight depends on

the composition of the gas as well as on the state of ionization of each species.

The level of ionization enters because free electrons must be included in the

average mass per particle m. When the gas is either completely neutral or

completely ionized,the calculation simplifies significantly.

Further investigation of the ideal gas law shows that it is also possible to

combine Eq. (2.11) with the pressure integral (2.9) to find the average kinetic

energy per particle. Equating, we see that

1

n

∫ +∞

0

nvv2dv =

3kbT

m(2.13)

However, the left-hand side of this expression is just the integral average

of v2 weighted by the Maxwell-Boltzmann distribution function. Thus we can

obtain the classical relation between the kinetic energy and the temperature

1

2mv2 =

3

2kbT (2.14)

It is worth noting that the factor of 3 arose from averaging particle velocities

over the three coordinate directions (of degrees of freedom). Thus the average

kinetic energy of a particle is 12kbT .

21

2.3.2 Quantum statistical mechanics

As has already been mentioned, there are stellar environments where the as-

sumptions of the ideal gas law do not hold even approximately. For instance,

in the pressure integral it was assumed that the upper limit of integration for

velocity was infinity. Of course, this cannot be the case since, from Einstein’s

theory of special relativity, the maximum possible value of velocity is c, the

speed of light. Furthermore, the effects of quantum mechanics were also ne-

glected in the derivation of the ideal gas law. When the Heisenberg uncertainty

principle and the Pauli exclusion principle are considered, a distribution func-

tion considers these important principles and leads to a very different pressure

equation of state when applied to extremely dense matter such as that found

in white dwarf stars and neutron stars. These exotic object will be discussed

in detailed soon because electron and neutron are fermions and to describe

them Fermi-Dirac distribution is needed.

Another statistical distribution function is obtained if it is assumed that

the presence of some particles in a particular state enhances the likelihood

of others being in the same state, an effect somewhat opposite to that of the

Pauli exclusion principle. Bose-Einstein statistics has a variety of applications,

including understanding the behavior of photons. Particles that obey Bose-

Einstein statistics are known as bosons.

Just as special relativity and quantum mechanics must give classical results

in the appropriate limits, Fermi-Dirac and Bose- Einstein statistics also ap-

proach the classical regime at sufficiently low densities and velocities. In these

limits both distribution functions become indistinguishable from the classical

Maxwell-Boltzmann distribution function.

Because photons possess momentum pγ = hνc, they are capable of delivering

an impulse to other particles during absorption or reflection. Consequently,

electromagnetic radiation results in another form of pressure. Substituting the

speed of light for the velocity v, using the expression for photon momentum

and using an identity for the distribution function, npdp = nνdν, the general

pressure integral 2.8, now describes the effect of radiation, giving

22

Prad =1

3

∫ +∞

0

hνnνdν (2.15)

At this point, the problem again reduces to finding an appropriate ex-

pression for nνdν. Since photons are bosons, the Bose-Einstein distribution

function would apply. However, the problem may also be solved by realizing

that nνdν represents the number density of photons having frequencies lying

in the range between ν and ν + dν. Multiplying by the energy of each photon

in that range would then give the energy density over the frequency interval

or

Prad =1

3

∫ +∞

0

uνdν (2.16)

where uνdν = hνnνdν. But the energy density distribution function is

found from the Plank function for blackbody radiation

uνdν =8πhν3

c31

ehνkbT − 1

dν (2.17)

Substituting this into Eq. (2.16) and performing the integration lead to

Prad =1

3aT 4 (2.18)

where a is the radiation constant found to be a = 4σc

= 7.56× 10−16 J/m3K4.

In many astrophysical situations the pressure due to photons can actually

exceed by a significant amount the pressure produced by the gas. In fact it is

possible that the magnitude of the force due to radiation pressure can become

sufficiently great that it surpasses the gravitational force, resulting in an overall

expansion of the system. Finally combining both the ideal gas and radiation

pressure terms, the total pressure becomes

Ptot =ρkbT

µmH

+1

3aT 4 (2.19)

23

24

Chapter 3

White dwarfs

Figure 3.1: The white dwarf, Sirius B,

beside the overexposed image of Sirius

A

In 1838 Friedrich Wilhelm Bessel 1

used the technique of stellar paral-

lax to find the distance to the star 61

Cygni. Following this first successful

measurement of a stellar distance,

Bessel applied his technique to an-

other likely candidate: Sirius (α Ca-

nis Majoris) that is the brightest star

in the Earth’s night sky with an ap-

parent magnitude of −1.46. In 1844

the German astronomer Friedrich

Bessel deduced from changes in the

proper motion of Sirius that it had

an unseen companion. Nearly two

decades later, on January 31, 1862,

American telescope-maker and as-

tronomer Alvan Graham Clark first observed the faint companion, which is

now called Sirius B. The detailed of their orbits about their center of mass

revealed that Sirius A and Sirius B have masses of about 2.3 M� and 1.0 M�,

1Friedrich Wilhelm Bessel (22 July 1784 – 17 March 1846) was a German astronomer,mathematician, physicist and geodesist. He was the first astronomer who determined reliablevalues for the distance from the sun to another star by the method of parallax.

25

respectively.

Clark’s discovery of Sirius B was made near the opportune time of apas-

tron, when the two stars were most widely separated (about 0°0′10′′). The

great difference in their luminosities (LA = 23.5 L� and LB = 0.03 L�) makes

observations at other times much more difficult.2

When the next apastron arrived 50 years later, spectroscopists had devel-

oped the tools to measure the star’s surface temperatures. From the faint

appearance, astronomers expected it to be cool and red but observations re-

vealed that Sirius B is a hot, blue-white star that emits much of its energy

in the ultraviolet. A modern value of the temperature of Sirius B is 27 000 K,

much hotter than Sirius A (9910 K).

The implications for the star’s physical characteristics were amazing. Using

the Stefan-Boltzmann law, 1.3, to calculate the size of Sirius B results in a

radius of only 5.5× 106 m ≈ 0.008 R�3. Sirius B has the same mass of the

Sun confined within a volume smaller than Earth. The average density of

Sirius B is 3× 109 Kg/m3, and the acceleration due to gravity at its surface

is about 4.6× 106 m/s2. On Earth, the pull of gravity on a teaspoon of white

dwarf material would be over 16 tons.

Now, if the white dwarf is the hot core of a star in which there are no nu-

clear reactions anymore; what can support a white dwarf against the relentless

pull of its gravity? The answer was discovered in 1926 by the British physi-

cist Sir Ralph Howard Fowler (1889-1944), who applied the new idea of the

Pauli exclusion principle to the electrons within the white dwarf. The qualita-

tive argument that follows elucidates the fundamental physics of the electron

2The solar luminosity is a unit of radiant flux (power emitted in the form of photons)conventionally used by astronomers to measure the luminosity of stars. One solar luminosityis equal to the current accepted luminosity of the Sun, which is 3.846×1026 W. This doesnot include the solar neutrino luminosity, which would add 0.023L�. The Sun is a weaklyvariable star, and its luminosity therefore fluctuates. The major fluctuation is the eleven-year solar cycle (sunspot cycle), which causes a periodic variation of about ±0.1%. Anyother variation over the last 200–300 years is thought to be much smaller than this.

3Solar radius is a unit of distance used to express the size of stars in astronomy equalto the current radius of the Sun. The solar radius is approximately 6.955× 108 m (about110 times the radius of the Earth, or 10 times the average radius of Jupiter). The SOHOspacecraft was used to measure the radius of the Sun by timing transits of Mercury acrossthe surface during 2003 and 2006. The result was a measured radius of (696342± 65)Km

26

Figure 3.2: Fraction of states of energy ε occupied by fermions. For T = 0, allfermions have ε ≤ εF , but for T > 0, some fermions have energies in excess ofthe Fermi energy.

degeneracy pressure described by Fowler.

3.1 Electron degeneracy

In an everyday gas at standard temperature and pressure, only one of every 107

quantum states is occupied by a gas particle, and the limitations imposed by

the Pauli exclusion principle become insignificant. Ordinary gas has a thermal

pressure that is related to its temperature by the ideal gas law. However, as

energy is removed from the gas and its temperature falls, an increasingly large

fraction of the particles are allowed in each state; thus all the particles cannot

crowd into the ground state. Instead, as the temperature of the gas is lowered,

the fermion will fill up the lowest available unoccupied states, starting with the

ground state, and then successively occupy the excited states with the lowest

energy. Even in the limit T → 0 K, the vigorous motion of the fermions in

excited states produces a pressure in the fermion gas. At zero temperature, all

of the lower energy states and none of the higher energy state are occupied.

Such aq fermion gas is said to be completely degenerate.

The maximum energy εF of any electron in a completely degenerate gas

at T = 0 K is know as the Fermi energy ; see Fig. (3.1). So, as in the Som-

27

merfeld model of conduction in metal, we have to solve the time independent

Schrödinger equation with an appropriate boundary condition. If we imagine

a three dimensional box of length L on each side and requires the vanishing

of the wave function, then this will lead to standing wave solutions. A more

satisfactory condition is imaging each face of the cube to be joined to the face

opposite it so that an electron coming to the surface is not reflected back in,

but leaves the cube, simultaneously reentering at the corresponding point on

the opposite it

ψ(x+ L, y, z) = ψ(x, y, z) (3.1)

ψ(x, y + L, z) = ψ(x, y, z) (3.2)

ψ(x, y, z + L) = ψ(x, y, z) (3.3)

The preceding equations are known as the Born-von Karman (or periodic)

boundary condition. The solution to the Schrödinger equation is always a wave

function of the form

ψk(r) =1√L3eik·r (3.4)

but thanks to the periodic boundary condition we can obtain only certain

discrete values of k. Obviously it is easier to express this condition in terms

of wavelengths:

λx =2L

kx, λy =

2L

ky, λz =

2L

kz(3.5)

where kx; ky e kz are integer quantum numbers associated with each di-

mension. Recalling that the de Broglie wavelength is related to momentum,

one gets

px =hkx2L

, py =hky2L

, pz =hkz2L

(3.6)

Now, the total kinetic energy of a particle can be written as

28

ε =p2

2m(3.7)

where p2 = p2x + p2y + p2z. Thus,

ε =h2

8mL2(k2x + k2y + k2z) =

h2k2

8mL2(3.8)

where k2 = k2x + k2y + k2z , is the analogous to the "distance" from the origin

in "k-space" to the point (kx, ky, kz). The total number of electrons in the

gas corresponds to the total number of unique quantum numbers, nx, ny and

nz times two, that arises from the fact that electrons are spin 12particles, so

ms = ±12implies that two electrons can have the same combination of kx,

ky and kz and still posses a unique set of four quantum numbers (including

spin). Now, each integer coordinate in k-space corresponds to the quantum

state of two electrons. When N is enormous, the occupied region will be

indistinguishable from a sphere of radius k =√k2x + k2y + k2z , but only for the

positive octant of k-space where all these integers are positive. This means

that the total number of electrons will be

Ne = 2

(1

8

)(4

3πk3)

(3.9)

Solving for k yields

k =

(3Ne

π

) 13

(3.10)

Substituting into Eq. (3.8) and simplifying, we find the Fermi energy is

given by

εF =2

2m(3π2n)

23 (3.11)

where m is the mass of the electron and n = NeL3 is the number of electrons

per unit volume. The average energy per electron at zero temperature is 35εF .

29

At any temperature above absolute zero, some of the states with an energy

less than εF will become vacant as fermions use their thermal energy to occupy

other, more energetic states. Although the degeneracy will not be precisely

complete when T > 0 K, the assumption of complete degeneracy is a good

approximation at the densities encountered in the interior of a white dwarf.

All but the most energetic particles will have an energy less than the Fermi

energy. To understand how the degree of degeneracy depends on both the

temperature and the density of the white dwarf, we first express the Fermi

energy in terms of the density of the electron gas. For full ionization, the

number of electrons per unit volume is

ne =

(Z

A

)ρ

mH

(3.12)

where Z and A are the number of protons and nucleons, respectively in the

white dwarf’s nuclei, and mH is the mass of a hydrogen atom. Thus the Fermi

energy is given by

εF =~2

2me

[3π2

(Z

A

)ρ

mH

] 23

(3.13)

The last result allows to obtain a condition for degeneracy by comparing

the Fermi energy with the average thermal energy of an electron ( 32kbT ): If

32kbT < εF , then an average electron will be unable to make a transition to

an unoccupied state, and the electron gas will be degenerate. That is, for a

degenerate gas we obtain,

T

ρ2/3<

~2

3mekb

[3π2

mH

(Z

A

)] 23

= D (3.14)

for Z/A = 0.5 the value of D is 1261 Km2/Kg2/3. When the quantity Tρ2/3

is

less than D electron degeneracy is quite weak and the Pauli exclusion principle

plays a very minor role to justify the usage of Maxwell- Boltzmann distribution;

when it is greater than D the complete degeneracy is a valid assumption.

30

3.2 Electron degeneracy pressure

We now estimate the electron degeneracy pressure by combining two key ideas

of quantum mechanics:

1. The Pauli exclusion principle, which allows at most one electron in each

quantum state;

2. Heisenberg’s uncertainty principle in the form of ∆x∆px ≈ ~.

When we make the unrealistic assumption that all of the electrons have the

same momentum p, Eq. (2.8) becomes

P ≈ 1

3nepv (3.15)

In a completely degenerate electron gas, the electrons are packed as tightly

as possible, and for uniform number density of ne, the separation between

neighboring electrons is about n−13

e . However, to satisfy the Pauli exclusion

principle, the electrons must maintain their identities as different particles.

That is, the uncertainty in their positions cannot be larger than their physical

separation. Identifying ∆x ≈ n− 1

3e for the limiting case of complete degeneracy,

we can use Heisenberg’s uncertainty relation to estimate the momentum of an

electron. In one coordinate direction, we get

px ≈ ∆px ≈~

∆x≈ ~n

13e (3.16)

However, in a three dimensional gas each of the directions is equally likely,

implying that < p2x >=< p2y >=< p2z > which is just a statement of the

equipartition of energy among all the coordinate directions. Therefore using

Eq. (3.12) one obtains

p ≈√

3~[(

Z

A

)ρ

mH

] 13

(3.17)

and for non relativistic electrons, the speed is

31

v =p

me

≈√

3~me

[(Z

A

)ρ

mH

] 13

(3.18)

Inserting equations (3.12), (3.16) and (3.18) into Eq. (3.15) for the electron

degeneracy pressure turns out to be

P ≈ ~2

me

[(Z

A

)ρ

mH

] 53

(3.19)

This is roughly a factor of two smaller than the exact expression for the

pressure due to a completely degenerate, non relativistic electron gas P

P =(3π2)

23

5

~2

me

[(Z

A

)ρ

mH

] 53

(3.20)

To appreciate the effect of relativity on the stability on the stability of a

white dwarf, recall that the previous equation (which is valid only far approx-

imately ρ < 109Kg/m3) is of the polytropic form P = Kρ5/3, where K is a

constant. This means that the white dwarf is dynamically stable. If it suffers

a small perturbation, it will return to its equilibrium structure instead of col-

lapsing. However, in the extreme relativistic limit, the electron speed v = c

must be used, instead of Eq. (3.18), to find the electron degeneracy pressure.

The result is

P =(3π2)

13

4~c[(

Z

A

)ρ

mH

] 43

(3.21)

In this case we find a situation of dynamical instability: the smallest de-

parture from equilibrium will cause the white dwarf to collapse as electron

degeneracy pressure fails 4. When the white dwarf is not stable anymore the

core collapses creating a supernova.

4In fact, the strong gravity of the white dwarf, as described by Einstein’s general theoryof relativity, act to raise the critical value of the exponent of ρ for dynamical instabilityslightly above 4/3.

32

3.3 Chandrasekhar limit

The requirement that degenerate electron pressure must support a white dwarf

star has profound implications. In 1931, at the age of 21, the Indian physicist

Subrahmanyan Chandrasekhar announced his discovery that there is a max-

imum mass for white dwarf. It this section we will consider the physics that

leads to this amazing conclusion. To calculate this limit we have to recall Eq.

(2.5) for the hydrostatic equilibrium. Remembering the definition of density

we can rewrite the hydrostatic equilibrium as

dP

dr= −GMwdρ

r2= −4

3πGρ2r (3.22)

where we indicated with Mwd the mass of the white dwarf. Integrating

Eq. (3.229 between any position r and the radius of the white dwarf Rwd and

using boundary condition that P = 0 at the surface to obtain a pressure as a

function of r

P (r) =2

3πGρ2(R2 − r2) (3.23)

3.3.1 The Mass-Volume relation

The relation between the radius, Rwd, of a white dwarf and its mass, Mwd,

may be found by setting the central pressure (Eq. (3.23) with r = 0), equal to

the electron degeneracy pressure, Eq. (3.21)

2

3πGρ2R2 =

(3π2)23

5

~2

me

[(Z

A

)ρ

mH

] 53

(3.24)

Then using the definition of the density and the volume of the sphere

(assuming constant density), this leads to an estimation of the radius of the

white dwarf:

Rwd ≈(18π)

23

10

~2

GmeM1/3wd

[(Z

A

)ρ

mH

] 53

(3.25)

For a 1 M� carbon-oxygen white dwarf, R ≈ 2.9× 106 m, too small by

33

Figure 3.3: Mass-radius relation for white dwarfs in both relativistic and non-relativistic case. As we can see white dwarf in non-relativistic case is alwaysstable and can’t collapse, while in the relativistic case a white dwarf can’t existover the Chandrasekhar limit.

roughly a factor of two but an acceptable estimate. More important is the

surprising implication that

MwdR3wd = MwdVwd = constant (3.26)

The volume of a white dwarf is inversely proportional to its mass, so more

massive white dwarfs are actually smaller. This mass-volume relation is a

result of the star deriving its support from electron degeneracy pressure. The

electrons must be more closely confined to generate the larger degeneracy

pressure required to support a more massive star. In fact, the mass-volume

relation implies that ρ ∝M2wd.

According to the mass-volume relation, piling more and more mass onto a

white dwarf would eventually result in shrinking the star down to zero volume

as its mass becomes infinite. However, if the density exceeds about 109Kg/m3,

there is a departure from this relation. To see why this is so, use 3.18 to

estimate the speed of the electrons: with a mass of about 2 M� the star would

be so small and dense that their electrons would exceed the limiting value of the

34

speed of light. This impossibility points out the danger of ignoring the effects

of relativity in our expressions for the electron speed and pressure. Because the

electrons are moving more slowly that the nonrelativistic 3.18 would indicate,

there is less electrons pressure available to support the star. Thus a massive

white dwarf is smaller that predicted; in other words, there is a limit to the

amount of matter that can be supported by electron degeneracy pressure.

3.3.2 Estimating the Chandrasekhar limit

An approximate value for the maximum white dwarf mass may be obtained

by setting the estimate of the central pressure (Eq. (3.23) with r = 0) with

ρ = Mwd/43πR3

wd, equal to Eq. (3.21) with Z/A = 0.5. The radius of the white

dwarf cancels, leaving

MCh ∼3√

2π

8

(~cG

) 32[(

Z

A

)1

mH

]2= 0.44 M� (3.27)

for the greatest possible mass. Note that Eq. (3.27) contains three fun-

damental constants ~, c and G; representing the combine effect of quan-

tum mechanics, relativity, and Newtonian gravitation on the structure of a

white dwarf. A precise derivation with Z/A = 0.5 results in a value of

MCh = 1.44 M�, called the Chandrasekhar limit. Sec. (3.3.1) shows the mass-

radius relation for white dwarfs. No white dwarf has been discovered with a

mass exceeding the Chandrasekhar limit.

It is important to emphasize that neither the nonrelativistic nor the rela-

tivistic formula for the electron degeneracy pressure developed here contains

the temperature. Unlike the gas pressure of the ideal gas law and the expres-

sion for radiation pressure, the pressure of a completely degenerate electron

gas is independent of its temperature. This has the effect of decoupling the

mechanical structure of the star from its thermal properties. However, the

decoupling is never perfect since T > 0. As a result, the correct expression for

the pressure involves treating the gas as partially degenerate and relativistic,

but with v < c. This is a challenging equation of state to deal with properly.

35

Figure 3.4: The Crab Nebula (catalog designations M1, NGC 1952, TaurusA) is a supernova remnant and pulsar wind nebula in the constellation ofTaurus. Correspond to a bright supernova recorded by Chinese astronomersin 1054 and is at a distance of about 2 Kp (about 6500 ly) from Earth. It hasa diameter of 3.4 p (11 ly), and is expanding at a rate of about 1500 Km/s. Atthe center of the nebula lies the Crab Pulsar, a neutron star of about 30 Kmacross with a spin rate of 30.2 Hz which emits pulses of radiation from gammarays to radio waves.

36

Chapter 4

Neutron stars

Two years after James Chadwick 1 discovered the neutron in 1932, a German

astronomer Walter Baade (1893-1960) and a Swiss astrophysicist Fritz Zwicky

(1898-1974) of Mount Wilson Observatory, proposed the existence of neutron

stars. These two astronomers, who also coined the term supernova, went on

to suggest that "supernovae represent the transition from ordinary stars into

neutron stars, which in their final stages consist of extremely closely packed

neutrons".

4.1 Neutron degeneracy

Because neutron stars are formed when the degenerate core of an old supergiant

star nears the Chandrasekhar limit and collapses, we take MCh for a typical

neutron star mass. A 1.4 M� neutron star would consist of 1.4M�mn

≈ 1057

neutrons, in effect, a huge nucleus with a mass number of A ≈ 1057 that is held

together by gravity and supported by neutron degeneracy pressure. In fact,

like electrons, neutrons are fermions and so are subject to the Pauli exclusion

principle. Recalling the same equation used for the electron degeneracy we

1Sir James Chadwick (20 October 1891 – 24 July 1974) was an English physicist who wasawarded the 1935 Nobel Prize in Physics for his discovery of the neutron in 1932. In 1941,he wrote the final draft of the MAUD Report, which inspired the U.S. government to beginserious atomic bomb research efforts. He was the head of the British team that worked onthe Manhattan Project during the Second World War. He was knighted in England in 1945for his achievements in physics.

37

obtain the radius-mass relation for a neutron star that might appear like

Rns ≈(18π)

23

10

~2

GM1/3ns

(1

mH

) 83

(4.1)

For Mns = 1.4 M�, this yields a value of 4400 m. As we found with Eq.

(3.25) for white dwarfs, this estimation is too small by a factor of about 3.

That is, the actual radius of a 1.4 M� neutron star lies roughly between 10

and 15 Km. As will be seen, there are many uncertainties involved in the

construction of a model neutron star.

This incredibly compact stellar remnant would have an average density of

6.67× 1017 Kg/m3, greater than the typical density of an atomic nucleus that

is 2.3× 1017 Kg/m3. In some sense, the neutrons in a neutron star must be

"touching" one another. At the density of a neutron star, all of Earth’s human

inhabitants could be crowded into a cube 1.5 cm on each side.

The pull of gravity at the neutron star is fierce. For a 1.4 M� neutron star

with a radius of 10 Km, g = 1.86× 1012 m/s2, about 200 billion times stronger

that the acceleration of gravity at Earth’s surface. An object dropped from a

height of one meter would arrive at the star’s surface with a speed of about

500 000 Km/h.

Another extremely important fact is the inadequacy of using Newtonian

mechanics to describe neutron stars that can be demonstrated by calculating

the escape velocity at the surface:

vesc =

√2GMns

Rns

= 1.93× 108 m/s = 0.643c (4.2)

Clearly, the effects of relativity must be included for an accurate descrip-

tion of a neutron star. This applies not only to Einstein’s theory of special

relativity but also to his general theory of relativity. Nevertheless, we will use

both relativistic formulas and the more familiar newtonian physics to reach

qualitatively correct conclusion about neutron stars.

38

4.2 Equation of state

To appreciate the exotic nature of the material constituting a neutron star

and the difficulties involved in calculating the equation of state, imagine com-

pressing the mixture of iron nuclei and degenerate electrons that make up an

iron white dwarf ate the center of a massive supergiant star. Specifically, we

are interested in the equilibrium configuration of 1057 nucleons, together with

enough free electrons to provide zero net charge. The equilibrium arrangement

is the one that involves the least energy.

Initially, at low densities the nucleons are found in iron nuclei. This is the

outcome of the minimum-energy compromise between the repulsive Coulomb

force, among the protons and the attractive nuclear force among all of the

nucleons. However, as mentioned in the discussion of the Chandrasekhar limit,

when ρ ≈ 109Kg/m3 the electrons become relativistic. Soon thereafter, the

minimum-energy arrangement of protons and neutrons changes because the

energetic electrons can convert protons in the iron nuclei into neutrons by

process of electron capture

p+ + e− −−→ n + νe (4.3)

Because the neutron mass is slightly greater than the sum of the proton

and electrons masses, and the neutrino’s rest-mass energy is negligible, the

electrons must supply the kinetic energy to make up the difference in energy;

mnc2 −mpc

2 −mec2 = 0.78 MeV.

We will obtain an estimation of the density at which the process of elec-

tron capture begins for a simple mixture of hydrogen nuclei (protons) and

relativistic degenerate electrons,

p+ + e− −−→ n + νe (4.4)

In the limiting case when the neutrino carries away no energy, we can

equate the relativistic expression for the electron kinetic energy to the differ-

39

ence between the neutron rest energy and combined proton and electron rest

energies and write

mec2

1√1− v2

c2

− 1

= (mn −mp −me)c2 (4.5)

or

(me

mn −mp

)2

= 1− v2

c2(4.6)

Although Eq. (3.18) for the electron speed is strictly valid only for nonrel-

ativistic electrons, it is accurate enough to be used in this estimate. Inserting

this expression for v leads to

(me

mn −mp

)2

≈ 1− ~2

m2ec

2

[(Z

A

)ρ

mH

] 23

(4.7)

Solving for ρ, one gets that the density at which electron capture begins is

approximately

ρ ≈ AmH

Z

(mec

~

)3 [1−

(me

mn −mp

)2] 3

2

≈ 2.3× 1010 Kg/m3 (4.8)

using A/Z = 1 for hydrogen. This is in reasonable agreement with the

actual value of ρ = 1.2× 1010 Kg/m3.

We considered free protons in the calculation above to avoid the complication

that arise when they are bound in heavy nuclei. A careful calculation that

takes into account the surrounding nuclei and relativistic degenerate electrons,

as well as the complexities of nuclear physics, reveals that the density must

exceed 1012Kg/m3 for the protons in 5626Fe nuclei to capture electrons. At

still higher densities, the most stable arrangement of nucleons is one where

the neutrons and protons are found in a lattice of increasingly neutron-rich

nuclei due to Coulomb repulsion between protons. This process is known as

40

neutronization and produces a sequence of nuclei such as 5626Fe, 52

28Ni, 6428Ni,

. . . , 11836 Kr. Ordinarily, there supernumerary neutrons would revert to protons

via the standard β-decay process

n −−→ p+ + e− + νe (4.9)

However, under the condition of complete electron degeneracy, there are

no vacant states available for an emitted electron to occupy, so the neutrons

cannot decay back into protons2.

When the density reaches about 4× 1014 Kg/m3, the minimum-energy ar-

rangement is one in which some of the neutrons are found outside the nuclei.

The appearance of these free neutrons is called neutron drip and marks the

start of a three-component mixture of a lattice of neutron-rich nuclei, nonrel-

ativistic degenerate free neutrons, and relativistic degenerate electrons.

The fluid of free neutrons has the striking property that it has no viscosity.

This occurs because a spontaneous pairing of the degenerate neutrons has taken

place. The resulting combination of two fermions (the neutrons) is a boson and

so is not subject to the restrictions of the Pauli exclusion principle. Because

degenerate bosons can all crowd into the lowest energy state, the fluid of paired

neutrons can lose no energy. It is a superfluid that flows without resistance.

Any whirlpools or vortices in the fluid will continue to spin forever without

stopping.

2An isolated neutron decays into a proton in about 10.2min, the half-life for that process.

41

Transition density Degeneracy

(kg m−3) Composition pressure

iron nuclei,

nonrelativistic free electrons electron

≈ 1 × 109 electrons become relativistic

iron nuclei,

relativistic free electrons electron

≈ 1 × 1012 neutronization

neutron-rich nuclei,

relativistic free electrons electron

≈ 4 × 1014 neutron drip

neutron-rich nuclei,

free neutrons,

relativistic free electrons electron

≈ 4 × 1015 neutron degeneracy pressure dominates

neutron-rich nuclei,

superfluid free neutrons,

relativistic free electrons neutron

≈ 2 × 1017 nuclei dissolve

superfluid free neutrons,

superconducting free protons,

relativistic free electrons neutron

≈ 4 × 1017 pion production

superfluid free neutrons,

superconducting free protons,

relativistic free electrons

other elementary particles (pions, ...?) neutron

As the density increases further, the number of free neutrons increases as

the number of electrons declines. The neutrons degeneracy pressure exceeds

the electron degeneracy pressure when the density reaches roughly 4× 1015 Kg/m3.

As the density approaches ρnuc, the nuclei effectively dissolve as the distinc-

42

tion between neutrons inside and outside of nuclei becomes meaningless. This

results is a fluid mixture of free neutrons, protons, and electrons dominated

by neutron degeneracy pressure, with both the neutrons and protons paired

to form superfluids. The fluid of pairs of positively charged protons is also

superconducting, with zero electrical resistance. As the density increases fur-

ther, the ratio of neutrons : protons : electrons approaches a limiting value

of 8 : 1 : 1, as determined by the balance between the competing precesses of

electron capture and β-decay inhibited by the presence of degenerate electrons.

The properties of the neutron star material when ρ > ρnuc are still poorly

understood. A complete theoretical description of the behavior of a sea of free

neutrons interacting via the strong nuclear force in the presence of protons

and electrons is not available yet, and there is little experimental data on

the behavior of matter in this density range. A further complication is the

appearance of sub-nuclear particles such as pions (π) produced by the decay

of a neutron into a proton and a negatively charged pion

n −−→ p+ + π− (4.10)

which occurs spontaneously in neutron stars when ρ > 2ρnuc.

The first quantitative model of a neutron star was calculated by J. R.

Oppenheimer (1904-1967) and G. M. Volkoff (1914-2000) at Berkeley in 1939.

This model display some typical features:

1. The outer crust consists of heavy nuclei, in the form of either a fluid

"ocean" or a solid lattice, and relativistic degenerate electrons. Nearest

the surface, the nuclei are probably 5626Fe. At greater depth and density,

increasingly neutron-rich nuclei are encountered until neutron drip begins

at the bottom of the outer crust (where ρ ≈ 4× 1014 Kg/m3).

2. The inner crust consists of a three-part mixture of a lattice of nuclei

such as 11836 Kr, a superfluid of free neutrons, and relativistic degenerate

electrons. The bottom of the inner crust occurs where ρ ≈ ρnuc, and the

nuclei dissolve.

43

Figure 4.1: Pulsars appear to be spinning neutron stars with rotation axestilted to their magnetic fields. Energetic electrons and light pour out themagnetic poles, and as the star spins the beam of light is swept across the skylike a lighthouse beacon.

4.3 Pulsar

Several properties of neutron stars were anticipated before they were observed.

For example, neutron stars must rotate very rapidly, in fact the decrease in

radius would be so great that the conservation of angular momentum would

guarantee the formation of a rapidly rotating neutron star.

The scale of the collapse can be found from Eq. (3.25) and Eq. (4.1) for

the estimated radii of a white dwarf and neutron star if we assume that the

progenitor core is characteristic of a white dwarf composed entirely of iron.

We obtain a ratio of the radii

Rcore

Rns

≈ mn

me

(Z

A

) 53

= 512 (4.11)

where Z/A = 26/56 for iron has been used. Now apply the conservation of

angular momentum to the collapsing core (which is assumed here for simplicity

to lose no mass, so Mcore = Mwd = Mns). Treating each star as a sphere with

a moment of inertia of the form I = CMR2, we have 3

3The constant C is determined by the distribution of mass inside the star. For example,C = 2

5 for a uniform sphere. We assume that the progenitor core and neutron star haveabout the same value of C.

44

Iiωi = Ifωf

CMiR2iωi = CMfR

2fωf

ωf = ωi

(Ri

Rf

)2

(4.12)

In terms of the rotating period T , this is

Tf = Ti

(Rf

Ri

)2

(4.13)

For the specific case of an iron core collapsing to form a neutron star

Tns ≈ 3.8× 10−6Tcore (4.14)

The question of how fast the progenitor core may be rotating is difficult an-

swer. As a star evolves, its contracting core is not completely isolated from the

surrounding envelope, so one cannot use the simple approach to conservation

of angular momentum described above. Anyway now we know that neutron

stars will be rotating very rapidly when they are formed, with rotation periods

on the order of a few milliseconds.

Jocelyn Bell and her Ph.D. thesis advisor, Anthony Hewish, spent two years

setting up a forest of 2048 radio dipole antennae over four and a half acres

of English countryside. They were using this radio telescope, tuned to a fre-

quency of 81.5 MHz, to study the scintillation that is observed when the radio

waves from distant sources known as quasars pass through the solar wind.

In July 1967 found a radio an object that emitted radio waves in regular in-

tervals: Hewish, Bell and their colleagues announced the discovery of these

mysterious "pulsating radio star" called pulsar, and several more were quickly

found by other radio observations. All known about pulsars share the following

characteristics, which are crucial clues to their physical nature:

45

• Most pulsars have periods between 0.25 and 2 s. with an average time

between pulses of about 0.795 s. The pulsar with the longest known

period is PSR 1841-0456 with T = 11.8 s and PSR J1748-2446ad is the

fastest (T = 0.001 39 s.

• Pulsars have extremely well defined pulse periods and would make ex-

ceptionally accurate clocks. For example, the period of PSR 1937-214

has been determined to be P = 0.001 557 806 448 872 75 s a measurement

that challenges the accuracy of the best atomic clocks. Such precise

determinations are possible because of the enormous number of pulsar

measurements that can be made, given their very short periods.

• The periods of all pulsars increase very gradually as the pulses slow down,

the rate of increase being given by the period derivative T ≈ 10−15 and

the characteristic lifetime is about 107 years.

In 1968 scientists discovered a pulsar associated with the Vela and Crab

supernovae remnants so they concluded that pulsars are rapidly rotating neu-

tron stars. In addition, the Crab pulsar PSR 0531-21 has a very short pulse

period of only 0.0333 s. No white dwarf could rotate 30 times per second with-

out disintegrating, and the last doubts about the identity of pulsars were laid

to rest.

4.4 Magnetars

Another property predicted for neutron stars is that they should have ex-

tremely strong magnetic fields.The "freezing in" of magnetic field lines in a

conducting fluid or gas implies that the magnetic flux through the surface of

a white dwarf will be conserved as it collapses to form neutron star. The flux

of a magnetic field through a surface S is defined as the surface integral

Φ =

∫S

B · dS (4.15)

46

where B is the magnetic field vector. In approximate temps, if we ignore

the geometry of the magnetic filed, this measn that the product of the magnetic

field strength and the area of the star’s surface remains constant. Thus

Bi4πR2i = Bf4πR

2f (4.16)

In order to use 4.16 to estimate the magnetic filed of a neutron star, we

must first know what the strength of the magnetic field is for the iron core of

a pre-supernova star. Although this is not at all clear, we can use the largest

observed white-dwarf magnetic field of B ≈ 5× 104 T as an extreme case,

which is large compared to a typical white-dwarf magnetic field of perhaps

10 T. and huge compared with the Sun’s global field of about 2× 10−4 T.

Then using 3.14, the magnetic field of the neutron star would be

Bns ≈ Bwd

(Rwd

Rns

)2

= 1.3× 1010 T (4.17)

This shows that neutron stars could be formed with extremely strong mag-

netic fields, although smaller values such as 108T or less are more typical.

Neutron stars with a such magnetic field (1011T) are called magnetars.

Magnetars’ magnetic fields are several orders of magnitude greater than typ-

ical pulsars and also have relatively slow rotation periods of 5 to 8 seconds.

Magnetars were first proposed to explain the soft gamma repeaters, objects

that emit bursts of hard X-rays and soft gamma-rays with energies of up to

100 keV. Only a few SGRs are known to exist in the Milky Way Galaxy, and

one has been detected in the Large Magellanic Cloud. Each of the SGRs is also

known to correlate with supernova remnants of fairly young age (∼ 104 years).

This would suggest that magnetars, if they are the source of the SGRs, are

short-lived phenomena. Perhaps the galaxy has many "extinct", or low-energy,

magnetars scattered through it.

The emission mechanism of intense X-rays from SGRs is through to be

associated with stresses in the magnetic fields of magnetars that cause the

surface of the neutron star to crack. The resulting readjustment of the surface

47

produces a super-Eddington release of energy (roughly 103 to 104 times the

Eddington luminosity limit in X-rays). In order to obtain such high luminosi-

ties, it is believed that the radiation must be confined; hence the need for very

high magnetic filed strengths.

Magnetars are distinguished from ordinary pulsars by the fact that the

energy of the magnetar’s field plays the major role in the energetics of the

system, rather that rotation, as is the core for pulsars. Clearly mush remains

to be learned about the exotic environment of rapidly rotating, degenerate

spheres with radii on the order of 10 Km and densities exceeding the density

of the nucleus of an atom.

48

Chapter 5

Conclusion

After some initial consideration about stellar evolution, the main target of this

thesis is the study of the easiest equation of state for a main sequence star,

white dwarfs and neutron stars in both relativistic and nonrelativistic case.

In particular we focused on the study of a Fermi degenerate gas and on how

the degeneration creates the pressure that supports and keeps stable a white

dwarf. Moreover we calculated the maximum mass (Chandrasekhar limit) that

can be supported by degeneration pressure. We concluded that for a correct

result we need to use relativistic equations.

As regards neutron stars we calculated that electrons are relativistic so,

it is necessary the use of special relativity and general relativity since the

density of this object is extremely high. In a neutron star the unique way

to sostain its mass is only neutron degeneracy pressure and, as in the case

of white dwarfs, exist a limiting mass for the stability of a neutron star, this

limit is called Tolman-Oppenheimer-Volkoff limit. If we consider the matter of

a neutron star made up of a great number of neutrons together with protons

and electrons in β equilibrium, then the Tolman-Oppenheimer-Volkoff limit is

about 2.8 M�.

Finally, the conclusive part of this thesis was dedicated to Magnetars, spe-

cial neutron stars characterized by extremely high magnetic fields, seeing the

equation that describes the evolution of the magnetic field when a white dwarf

collapses.

49

In spite of all the results we can get and the theories we can create, the

universe hides always something unexpected. From the first observations of

the sky to nowadays passed about 5,000 years: may be changed the way and

means for observation, but the amazement and wonder it inspires in all of us is

always the same. So as wrote the poet Sarah Williams: Though my soul may

set in darkness, it will rise in perfect light; I have loved the stars too fondly to

be fearful of the night.

50

Appendix A

Nuclear reactions

In nuclear physics and nuclear chemistry, a nuclear reaction is semantically

considered to be the process in which two nuclei, or else a nucleus of an atom

and a subatomic particle (such as a proton, neutron, or high energy electron)

from outside the atom, collide to produce one or more nuclides that are different

from the nuclide(s) that began the process. Thus, a nuclear reaction must cause

a transformation of at least one nuclide to another. If a nucleus interacts with

another nucleus or particle and they then separate without changing the nature

of any nuclide, the process is simply referred to as a type of nuclear scattering,

rather than a nuclear reaction.

In 1917, Ernest Rutherford was able to accomplish transmutation of nitro-

gen into oxygen at the University of Manchester, using alpha particles directed

at nitrogen

N + α −−→ O + p (A.1)

This was the first observation of an induced nuclear reaction, that is, a

reaction in which particles from one decay are used to transform another atomic

nucleus. Eventually, in 1932 at Cambridge University, a fully artificial nuclear

reaction and nuclear transmutation was achieved by Rutherford’s colleagues

John Cockcroft and Ernest Walton, who used artificially accelerated protons

against lithium-7, to split the nucleus into two alpha particles. The feat was

51

popularly known as "splitting the atom", although it was not the modern

nuclear fission reaction later discovered in heavy elements, in 1938.

In writing down the reaction equation, in a way analogous to a chemical

equation, one may in addition give the reaction energy on the right side

Targetnucleus + projectile −−→ Finalnucleus + ejectile + Q (A.2)

The reaction energy (the "Q-value") is positive for exothermic reactions

(It occurs spontaneously) and negative for endothermic reactions (It takes

place only with the input of energy from outside). On the one hand, it is

the difference between the sums of kinetic energies on the final side and on the

initial side. But on the other hand, it is also the difference between the nuclear

rest masses on the initial side and on the final side. As regards the exothermal

reactions the final total mass lesser than the initial total mass. This lost mass

become energy according to the equation 1.1.

Nuclear Fusion

We said that in the our Sun’s core takes place a very special nuclear reaction

in which two or more atomic nuclei come very close and then collide at a very

high speed and join to form a new type of atomic nucleus. A substantial energy

barrier of electrostatic forces must be overcome before fusion can occur. At

large distances, two naked nuclei repel one another because of the repulsive

electrostatic force between their positively charged protons. If two nuclei can

be brought close enough together, however, the electrostatic repulsion can be

overcome by the attractive nuclear force, which is stronger at close distances.

We easly understand that kinetic energy (and so the temperature) required for

this event is much higher of everything on the Earth. In a star the temperature

is of the order of 107 K and at this temperature this process become just

routine.

52

The first set of nuclear reactions that occur in a star is between two protons

(ionized nuclei of hydrogen) and are described by this equations:

H + H −−→ D + β + νe (A.3)

D + H −−→ He + γ (A.4)

He + He −−→ 2 H + He (A.5)

The net result of the cycle is the conversion of 4 hydrogen atoms in a helium

atom with the liberation of energy equal to 26.73 MeV. The γ photon emitted

during this first set of reactions exerts a radiation pressure on the outer layers

of the star, along with the pressure of stellar gas, it opposes the gravitational

contraction during the main sequence.

We have already said that when most of the hydrogen inside the star is

over, this will enter the red giant phase. During this phase, the core tempera-

ture reaches 108 K fueling the fusion of helium too according to the following

processes:

H + He −−→ Li (A.6)

He + He −−→ Be (A.7)

Be + He −−→ C (A.8)

Once the C is formed another series of thermonuclear reactions can take

place, the so-called carbon-nitrogen cycle:

53

Figure A.1: The proton-proton cycle in which four nuclei of hydrogen are trans-formed into an atom of helium, through two intermediate elements: deuteriumand helium 3.

C + H −−→ N + γ (A.9)

N −−→ C + β + νe (A.10)

C + H −−→ N + γ (A.11)

N + H −−→ O + γ (A.12)

O −−→ N + β + νe (A.13)

N + H −−→ C + He (A.14)

In that cycle carbon is involved only as a catalyst, reforming continuously.

All this process that create every element from the fusion of two hydrogen

atoms is called nucleosynthesis.

54

Bibliography

[1] A. E. Roy, D. Clarke - Astronomy principle and practice - IoP

[2] S. L. Shapiro, S. A. Teukolsky - Black holes, White dwarf and Neutron

stars the physics of compact object - Wiley interscience publication

[3] B. W. Carroll, D. A. Ostlie - Modern stellar astrophysics - Addison wesley

publication company

[4] K. Huang - Meccanica statistica - Zanichelli

[5] N.W. Ashcroft, N. D. Mermin - Solid state physics - Cengage learning emea

[6] Wikipedia, the free encyclopedia

[7] Appunti da internet

55