Post on 03-Aug-2020
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
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55