Download Introduction Introduction to to Astrophysics Astrophysics

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Astrophysics
Ronald L. Westra
Maastricht University
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1. Introduction
Astronomy is a fascinating and exciting field. For some it is a lifetime hobby, enjoyed from young
children to centenarians. For others it is their vocation and becomes their profession. The word
‘astronomy’ itself derives from the Greek aster meaning ‘star’, and nomos meaning ‘law’, and
originally referred to the mathematical laws governing the motion of the stars and planets. Astronomy
essentially is an observational science, and all astronomical theories are justified by their agreement
with astronomical observations. Until recently, observations were limited to optical telescopes or the
naked eye. Since some decades observations have extended to other parts of the electromagnetic
spectrum. More recently cosmical particles are directly studied, and at present some groups endeavor
to detect the illustrious gravitational waves predicted by Einstein’s General Relativity Theory.
Astrophysics is the science that uses physics to interpret astronomical events. As such, astrophysics is
a branch of both Astronomy and Physics. The field of astrophysics is now rapidly developing: each
year brings an increased number of significant and exciting discoveries based on data from space- and
ground-based observatories, spacecrafts, rockets and balloons. All this information has deepened and
broadened our understanding of the structure and history of the universe and its constituents.
Astronomy and astrophysics are vastly extensive areas, ranging from their historical development and
philosophic principles to highly specialized mathematical theories or experimental techniques. In this
course we will at most scratch its surface. We will thereby focus on three parts of astrophysics; stellar
evolution, stellar dynamics, and the evolution of the universe.
This course is self-contained in the sense that this syllabus should provide sufficient material for the
tasks and assignments. This course is designed for students with no prior knowledge of astronomy, but
makes moderate use of some facts you have learned in the preceding physics course. You do not need
to have your own telescope to follow this course. However, to let you share some of the fun of
astronomy, there will be an optional activity at an – as yet undecided – observatory and/or
planetarium, which may be of particular interest to students who do not normally have access to a
telescope and photographic equipment.
Ronald Westra,
February 27 2003, Maastricht
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2. Astronomic Scales in Space and Time
As we gaze out in a bright and starry night we wonder and ponder about the beauty and marvels of the
celestial sky. We see the moon, some planets, various stars, perhaps a meteor, perhaps the Milky Way.
How far are these celestial objects and what is their age? Numerous generations have asked these
questions, and only the last generations have started to offer the first answers. May be these answers
will become obsolete and ludicrous in future generations, as have so many of our earlier ‘theories’ of
the universe, maybe they will stay in place. They are our answers here and now. Let us make a voyage
through space and time – according to our present model of the universe.
2.1. A Voyage into Spatial Dimensions
We start our journey with a voyage in the dimensions of space. Let us start with our local measure, the
length of our body. The typical dimension of objects that surround us ranges from a few centimeters to
a few meters (we stick to SI-units). Let us first go down the scale with powers of 10. At 10-3 m we
encounter the typical components of a PC; resistors, condensators and transistors. Typical animal and
plant cells can be found 10-4 m. Downward from 10-6 m we find macromolecules, such as the
celebrated DNA. Their constituents, the atoms, we meet at 10-9 m. Going downward in scale it now
becomes very quiet. Zooming in on the atoms for many magnitudes (powers of 10) we experience
only a vast emptiness. Finally, beyond 10-15m we find the protons and neutrons, and downward we
find their constituents, the most elementary particles presently known; the leptons i.e. electrons and
quarks. Current theories belief these particles themselves to be built from the most essential building
blocks; the so-called strings. These are found at the so-called Planck-length of 10-36 m. Here we have
entered the realm of not-vindicated elementary physical theories, and our journey reverses. After
zooming out some 36 magnitudes we are back at our own level of experience in the world that
surrounds us. Going up, at 103 m we see the hills grow into mountains. At 106 m we detect the earth’s
continents, and at 107 m we see the entire earth with a diameter of 1.2.107 m.
Fig. 1. Most left:
starting of our
journey is earth.
Left: Image of our
sun in visible light.
Right: The largest
planet in the solar
system is Jupiter.
One magnitude further we see the earth-moon system with a diameter of 7.6.108 m. At 3.1011 m we
have the diameter of the earth’s orbit around the sun. The distance to the sun is often used as yardstick,
called 1 AU = Astronomical Unit = 1.5.1011 m. From the sun, we find the planets: Mercury, Venus,
Earth, Mars, the asteroid belt, Jupiter, Saturn, Neptune, Uranus, and finally, at 6.0.1012 m = 40 AU
distance from the sun we find the most distant planet Pluto (its status as planet nowadays questioned).
Light from the sun travels this distance in 6.0.1012 m /3.0.108 m/s ~ 20,000 seconds, i.e. sunlight needs
more than five hours to reach Pluto.
Somewhere beyond Pluto ends the realm of the sun at the heliopause, and we measure distances in
lightyears – the distance light travels in 1 year = 3.107 s µ 3.108 m/s = 9.46.1015 m. Astronomers prefer
the so-called parsec (pc) = 3.26 Lightyears. This is the distance of a star that virtually moves 1” at the
sky due to the annual movement of the earth around the sun.
Assignment 1: Explain this motion, and validate the correspondence of 1 pc = 3.26 lightyears.
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The typical distance between stars in our neighborhood of the galaxy is about 2 pc ~ 6 lightyears ~ 6.
1016 m. At 24 kpc ~ 8000 lightyears ~ 8.1019 m we are at the diameter of our galaxy, the Milky Way.
At large magnifications we observe that the universe is filled with hundreds of billions of galaxies.
According to their shape galaxies can be classified as spiral, elliptical or irregular. Our Milky Way is a
beautiful spiral galaxy. Our nearest large neighboring galaxy is the Andromeda-nebula (its historical
name – but it is a galaxy) at a distance of approximately 2 million lightyears from earth.
Fig. 2 The galaxy M31, known
as the Andromeda nebula
In fact the Milky Way and the Andromeda nebula are gravitationally bounded and form a couple. This
couple itself is part of a larger system of galaxies, called the local group. The diameter of the local
group is some six million lightyears. Most galaxies in the universe are not single but are part of larger
aggregations of galaxies, the so-called clusters. Some of these clusters again aggregate in so-called
super clusters. A typical super clusters contains dozens of individual clusters spread over a region of
space of some 100 million lyrs across.
Fig. 3. Collection of galaxies. The three fuzzy
galaxies in the lower left of the figure are in the
process of merging, resulting in huge veils of
stars accompanying them. The ‘small’ crisp
galaxy almost in the center is actually on the
background and is far more distant than this
collection.
At even larger scales the aggregation of galaxies form an intricate three-dimensional structure
resembling a sponge. Most matter is congregated in small filament-like structures, and matter is
separated by gigantic spherical voids. The size of these voids is roughly between 100 million to 400
million lyrs. This structure is caused by the gravitational pull of matter, it ‘rips’ the holes in
continuous space. In this sponge-structure is a remarkable formation called ‘the great wall’ or the
‘central attractor’. It is the largest structure in the known universe and exerts its gravitational pull on
all visible matter.
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Fig. 4. Large-scale map of the observable universe
showing the the largest structures visible in the
universe. Each point in this diagram represents one
single galaxy – that on its own consists of about
1011 stars. The prominent structure running
diagonally across the upper part has been named
the Great Wall. It extends for at least 750M
lightyears, and likely even more as it is on one end
obscured by dust in the plane of our galaxy, and on
the other end has not yet been mapped. It is less
than 23M lightyears thick. In the southern sky
there is a corresponding structure called the
Southern Wall. Because neither the Northern Wall
nor the Southern Wall have been mapped fully, it
may even be possible that they are part of one
much larger structure as they join together in the
parts of the sky that have not yet been examined.
Also visible is the ‘Swiss-cheese’ of the universe;
in between the galaxies are large spherical voids.
Our local group itself is heading towards the Great Wall. Overall, however, the universe is expanding
according to the law of Hubble: the velocity that two galaxies separate from each other increases
linearly1 with their distance. This causes the entire universe itself to expand. This expansion will be
discussed later, but we already notice that it is not an expansion in a void, but an expansion of space
itself.
Finally we find the entire universe. The size of the entire universe depends on your favorite
cosmological theory. Traditional big bang theories gives an upper estimate of age-of-the-universe µ
velocity-of-light ~ 14.109 years µ 3.107 second/year µ 3.108 m/s = 1.26.1026 m. According to the
inflation theory, the size is even bigger, and in various theories, including some string theories, our
universe is but the local and observable part of an otherwise infinite multiverse.
We have traveled 36 magnitudes down and 26 magnitudes up and found ourselves about in the middle.
The exact middle is found at about 10 km, the size of a small town like Maastricht. Is it a mere
coincidence that man is half-way this scale, or does this tell something about our observational
abilities, and will not observers at all scales find themselves stuck about in the middle?
2.2. A Travel in Time
Fig. 5. The Universe at
the young age of
300,000 years. The
colors represent
temperature fluctuations
in the Cosmic
Background Radiation
(courtesy: Wilkinson
Microwave Anisotropy
Probe)
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We continue our journey with a voyage
in the dimension of time. We start in the
distant past when the whole universe as
we know it started in one titanic
explosion called the ‘Big Bang’. If we
follow the big bang theory – and we
will, the universe started in one
spontaneous event, some 14 billion
years ago. It started as a mathematical
singularity, as it was infinitely dense
and infinitely small. In this singularity
our concepts like space and time had no
valid meaning. In the first split second
only in first order it is linear.
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after its beginning the universe grew from this absolute singularity to the size of several light years. In
the first phase the universe was extremely hot and opaque as mass and radiation were ‘coupled’. Only
after 300,000 years matter and radiation became decoupled, and the universe suddenly became
transparent. After one billion years the first proto-galaxies formed. This caused the first stars to shine
and thus the formation of the first heavy elements. This in its turn enabled the formation of more
extensive galaxies, including our own galaxy. The initial matter still predominantly H and a bit He.
Highly massive stars burned fast and
when exhausted they exploded as
Fig. 6 The Giant Impact
Theory suggests that a
colossal super novae, After about 10
Mars-sized object
billion years – 4.6 billion years ago –
crashed into the early
our solar system formed. The
Earth. Most of the debris
formation of earth-moon system
thrown into space fell
happened as the result of a primordial
back on Earth, but a
collision some 4.5 billion years ago . A
fraction aggregated into
proto-planet, about the size of Mars,
the Moon. This theory is
collided at high speed with the nearly
supported by the similar
fully formed Earth. The collision
composition of rocks on
shattered Earth, and pulverized the
the Earth and Moon.
(courtesy: BBC))
incoming planet. Most of the impactor
rained down on to, and became
incorporated into, the Earth. Some 10% of the mass was spread out into an incandescent disc around
the Earth - a scorching equivalent of Saturn's rings. It was out of this material that the Moon was
formed in a matter of decades. In the past 4 billion years Earth witnessed mostly periods of rest, in
which geological events like continental drift and evolution of live occurred. Only in the last few
thousand years Earth has experienced the presence of humans. Which brings us to the presence.
What may the future hold? In about some 5-6 billion our sun will have burned out, and grow to the
size of a red giant encapsulating the earth orbit – and thereby destroying earth – before it will explode
and become a rapidly spinning dense neutron star.
But even before that, in about 3 billion years from now, we will be visited aliens. Our nearest large
neighbor galaxy is the Andromeda-nebula (M31) – see figure 2. It is heading towards us with a
velocity of 120 km/sec and will collide with the milky way in approximately three billion years. In this
violent event the central super-massive black holes of both galaxies will coalesce in a gargantuan
explosion. Most of the stars in both galaxies will be affected, either by being swung out in the extreme
emptiness of intergalactic space or by colliding to each other, and a large proportion of the stars will
be sucked down by the newly-formed super-massive central black hole of the new system.
Assignment 2: M31 is moving towards us relative to the Galactic center at a speed of
approximately 120 km/s. Its distance to earth is approximately 2 million lightyears. in
how many years from now will we collide based on these figures?
However, as this motion is accelerated due to gravitational interaction, the merger will be much
sooner. In about 3 billion years, the two galaxies will collide and then over about 1 billion years after a
very complex gravitational dance they will merge to form an elliptical galaxy2.
For even the more remote future, the prospects are not bright either. Either there is enough mass in the
universe to ultimately halt the expansion of space and let it fall back on itself in a ‘ Big Crunch’ in
many billions of years. Or there is not enough mass, and the universe keeps on expending until in
about some 1036 years all protons have decayed to gamma-photons and the total very very large
universe is totally empty of matter and only filled with radiation of ever lower frequencies. It is totally
dark and empty at absolute minimum temperature of 0 K.
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There is a beautiful galaxy-merger movie by Dr. John Dubinski at:
http://www.astro.soton.ac.uk/PH308/galaxies/mergers/MWmerge.mpg
which shows what happens when galaxies "collide".
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3. Stellar Evolution
3.1. The Sun
Our local star, the sun, is a typical main-sequence star of spectral type ‘G2V’. As such, it has no
unique claims to set it apart from the 1011 other main-sequence stars in our local Galaxy, the Milky
Way, or the perhaps 1020 other main-sequence stars in the observable universe. It is perfectly normal
for its type in terms of the usual stellar parameters. The only apparently remarkable aspect is that its
third planet has evolved a biology – including intelligent life, and we have no evidence whether that
aspect is unusual or not!
Assignment 1: How is it that we can classify our sun among the vastitude of stars, as the only
feature we can examine is the intensity-variations over their electromagnetic spectrum?
Table 1. Some characteristics of the sun.
radius (Rü)
mass (Mü)
mean density (rü)
total energy output (Lü)
age
core temperature
surface temperature
distance to earth
7 1010 cm
2 1033 g
1.4 g/cm3
3.82.1026 Joule/sec
1.5 1017 sec
5 106 K
5 103 K
1.5 1013 cm
If we set out to understand the stars, let us first study our own sun. Our sun is a massive rotating,
(almost) spherical body, consisting mostly of the elements H and He. The sun produces the vast
amount of 3.82.1026 Joule/sec of electromagnetic radiation in a process called nuclear fusion. The sun
is a subtle equilibrium between the explosive action of the nuclear fusion and the contracting pressure
of gravitation. These two actors, gravity and nuclear fusion, define the entire evolution of the sun. At
the center of the sun the gravitational forces are humongous. This results in extreme high pressures
and temperatures. Under these conditions all atoms are stripped of their electrons. This situation,
where matter consists of free nuclei and electrons, is called a plasma. This combination of high
pressure and temperatures acting on a plasma creates the perfect condition for the process of nuclear
fusion.
Assignment 2: What is the basic difference between nuclear fusion and nuclear fission, and
under what conditions will fusion prevail over fission?
In nuclear fusion four H nuclei join to form one He nucleus under emission of one energetic photon,
besides a neutrino and two electrons:
411 H → 42 He + 2 01 e + γ + ν + 26.7 MeV
This results in a high flux of powerful gamma-photons, neutrinos and electrons from the core of the
sun. However, in the higher layers of the sun the g-photons are immediately absorbed by the resident
H and He-nuclei. This absorption results in the heating of these layers, which in turn balances the
gravitational pressure. Eventually, the photon is re-emitted in a random direction. In all, this process of
absorption and emission generates a steady flux of photons and convective heat streams from the core
to the surface.
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Assignment 3: Argue how the combination of massive thermal convection and an ionized
plasma creates the ideal conditions for strong magnetic currents.
The average length an individual photon travels between emission and absorption is only 1 cm. Due to
this incessant process of absorption and emission the journey of one specific photon from the core to
the surface on average takes 800,000 year!
Assignment 4: How can we find out whether the sun has actually stopped central thermofusion
in the past 800,000 year?
Assignment 5: Calculate the average time a photon is absorbed, using that the radius of the sun
= 6.96.108 m.
3.2. General Stellar Parameters
Our excursion to the sun has provided us with the main mechanism for stellar equilibrium: gravity
versus nuclear fusion. The nuclear fusion is driven by gravitational pressure at the core and the ample
supply of ionized hydrogen. The gravitational pressure itself stems from the total mass of the star.
Thus, we come to two main parameters that define stellar types: 1. total mass and 2. chemical
composition. In practice, the latter means the ratio between H and He.
Assignment 6: Should not the age of a star be considered as a basic stellar parameter?
In the normal stellar equilibrium state huge amounts of hydrogen are transformed to helium.
Consequently, after some time the main supply of fuel for the thermofusion, hydrogen, is exhausted.
At that moment there is nothing that can halt the gravitational pull and the star implodes! We will
discuss this situation later. Now we consider how the life-expectancy of a star depends on the basic
stellar parameters mass and chemical composition. As we now understand the basic mechanism of
stellar equilibrium, we would expect the life-expectancy of a star to be proportional to its total mass:
the more hydrogen-fuel – the longer the fusion process lasts. The real situation, however, is directly
the reverse: the more massive a star – the shorter its lifetime. Hence: massive stars mean young stars.
Assignment 7: What does this fact mean for the ratio between thermonuclear energy
production and gravitational pressure as the mass of a star increases?
An observational phenomenon known for millennia is that stars differ in color. Some stars are blue,
others are red or green. This has led to the definition of the spectral type of a star. Depending on its
most dominant color, stars are classified to one of the following spectral types 3:
O–B–A–F–G–K–M–R–N–S
Here B stands for Blue, R for Red, G for Green. This classification denotes the spectral sequence from
Blue to Red as in a rainbow. In this classification there are detailed sub-divisions. For instance, our
sun is of spectral type ‘G2V’.
Assignment 8: Using Wien’s law we find that our sun has its optimum intensity in the visible
spectrum in the color green (for this reason it is a G2V-star, ‘G’ for ‘green’). Discuss from this
fact why evolution on earth has favored plants being green. What color should you design plants
near a B-spectral type star?
3
Some male students simply memorize this as: ‘Oh, Be A Fine Girl, Kiss Me Right Now – Ssssmack!’, whereas
some female students favor the G = ‘Guy’ or ‘Get-lost’ alternative.
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The basic principle behind the spectral type can be understood from the phenomenon of black body
radiation. If a black body is heated it starts to emit electromagnetic radiation. As the heating is
increased, at a certain moment a sufficient fraction of the electromagnetic radiation enters the visible
spectrum. As the heat increases, we will observe the black body as glowing from invisible infrared
through red, orange, yellow, green, blue, violet to invisible ultraviolet.
Assignment 9: Explain why in these latter stages we will experience the body as white.
The spectral type is so important because it can be directly observed. It is found to be directly related
to all kind of fundamental stellar characteristics, such as its chemical composition (from the emission
and absorption lines in the spectrum), surface temperature (using the relation between temperature and
dominant color as in black body radiation known as the wavelength-displacement law of Wien: lmax =
constant/T), absolute luminosity i.e. the cumulative energy over the entire spectrum (again using black
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body radiation, where the luminosity L relates to the surface temperature T as: Labs = constant µ T ).
Of course, on earth we measure the relative luminosity rather than the absolute luminosity. As stars
are on great distances from earth 4. Thus, the total light is uniformly distributed over a sphere as the
light spreads out in space. Thus the relation between absolute and relative luminosity is:
Lrel =
Labs
4πr 2
where r denotes the distance from the star to earth. For stars with a known distance to earth we can
thus estimate the absolute luminosity.
Assignment 10: Propose an observational method to measure the distance to at least some of
the visible stars.
As for many stars the absolute luminosity is not available, since the days of the Greek philosopher
Hipparchos astronomy uses the concept of the relative magnitude of a star. The relative magnitude m
of a star is a measure for the relative luminosity of a star, nowadays defined as:
m = −2.5 log Lrel
This relative magnitude is what we directly observe of a star. In the same way we define the absolute
magnitude M. The magnitude serves to describe the difference in observed luminosity between stars,
such as:
a Lyrae (Vega) with relative magnitude 0m.14 is 1.19 magnitudes brighter than a Cygni
(Deneb) with relative magnitude 1m.33.
Using color filters, the magnitude can also be used for specific parts of the electromagnetic spectrum.
In this way, we can define the ultraviolet magnitude: U = mU , the visual magnitude V = mV, and the
blue magnitude B = mB . Using these, we can – for instance – calculate the difference between
ultraviolet and blue magnitude of a star, U – B.
Assignment 11: Demonstrate that the difference U – B is independent of the distance from the
star to earth.
Early in the 20th century the astronomers Hertzsprung (Denmark) and Russel (USA) jointly designed a
diagram for the classification of stars that now bears their name: the Hertzsprung-Russel Diagram,
short HRD. Originally it plots the absolute magnitude M versus the spectral type for a number of
4
The star closest to the sun is Alpha Proxima Centauri at approximately 4.2 light years º 3.78.10
13
km.
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nearby stars. See figure 1. Later improvements included the difference between spectral magnitudes,
such as U – B, that are a measure for the spectral type and independent of the distance of the star. As
we argued above, the spectral type is a measure for the surface temperature, so we can consider the
HRD also as a schematic representation of the relation between surface temperature and total energy
output .i.e. luminosity.
3.3. Major Components in the Hertzsprung-Russel Diagram
Let us fill the HRD with data from stars with known (absolute) luminosity and spectral type. At one
glance we notice that most stars fall within a narrow band on the HRD. This band is called the main
sequence. It contains the majority of all stars, including our own sun. The existence of a narrow band
of main sequence stars indicates that for this prevalent type there exists a well-defined relation
between luminosity and surface temperature.
Next, we notice clusters in the upper-right and in the lower-left of the HRD. The upper-right cluster
contains the so-called giants, i.e. stars of gigantic masses compared with the sun. Below the main
sequence we find the dwarfs, small stars. Left the blue dwarfs, right the white dwarfs.
Fig. 1. Original Hertzsprung-Russell Diagram ( HRD)
3.7. Initial stages of Stellar Evolution
Distributed over the galaxy are huge clouds of dust and ice. The temperature is near the absolute
minimum of 0 K. These interstellar clouds are mainly composed of pure H, though all past super nova
contribute to some level of contamination with higher elements, see figure 8.
Assignment 12: Can you explain the presence of elements heavier than Fe in figure 8?
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Assignment 13: How could we estimate the age of the sun from contemporary observations of
the atmosphere of the sun?
These clouds act as star incubators, and they are the main sites for stellar formation. Convection in
these clouds can give rise to inhomogenities that can cause gravitational contraction. Such local
accumulations can act as seeds for further condensations. More and more matter is attracted to the
center. In combination with the conservation of angular momentum this leads to the formation of a fast
rotating accretion disk. In the convective whirls around this disk smaller entities may grow that can
eventually grow to planets. Depending on the masses involved, this may last 105 to 108 years. As the
core increases in mass, its central pressure and temperature increase until the point where thermal H
fusion commences. Then a shock wave passes through the cloud, signaling the birth of a star. The
bright radiation of the new star quickly (in astronomical terms) blows away all dust and smaller
particles, and soon after the accretion disk is driven away.
3.4. The Final Stages of Stellar Evolution
Now what happens when the amount of hydrogen in the central core of a star becomes exhausted? The
productivity of the nuclear fusion process will drop, and the generated heat and pressure will not
longer compensate the gravitational pressure. Hence, the star will start to contract. If the sun could not
counteract its own gravitational pull one can calculate that it would collapse in a time: Gρ where G
is Newton’s gravity constant; G ~ 6.7 10-8 cm3g-1sec-2, and r the average density (see table 1) 1.4
g/cm3. This leads to a collapse time of less than one hour!. During this contraction, however,
gravitational energy is transformed to heat. The plasma in the stellar core behaves like an ideal gas,
and therefore this heat would temporary raise the pressure and thus slow the contraction somewhat.
But as the heat permeates outwards, the star inevitably collapses. Is there nothing that can halt this
collapse? Indeed, there are other types of nuclear reactions that start at higher temperatures. At about
108 K Helium – now in ample supply because of the H-fusion – is fused with the remaining hydrogen
to Li (lithium):
He + H Ø Li + n
Thus, a new equilibrium state has been reached which can lasts several millions years – depending on
the remaining supply of hydrogen. During this equilibrium the star swells up to gigantic proportions
and becomes a red giant. For our sun this means that it would swell to the orbit of Mars, thus
engulfing the earth. Fortunately, this event lies about 5 billion years from us. As finally this resource
becomes exhausted, the collapse resumes and the star again starts to contract. This contraction
continues until the pressure and temperature is sufficiently raised for the next fusion process: He to C
(carbon)5.
342 He→126 C
This process of stable thermonuclear equilibria intermitted with gravitational contraction and heating
is repeated until the nuclear mass number of the produced fusion element reaches 56, see figure 2.
Figure 2 shows that the nuclear binding energy has its maximum at atom mass 56 which corresponds
to Fe (iron).
5
As the early universe was almost void of carbon, all carbon since has been produced in supernovae. Hence, we
all are made from stellar debris!
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Fig. 2. Binding energy in MeV per nucleon as function of mass number A.
Assignment 14: Argue from figure 2 how much energy can be gained from fusing two H nuclei
into 1 He nucleus. Moreover, argue how above mass number 56 nuclear fission can generate
energy.
Above mass number 56 no energy can be gained from nuclear fusion. At that moment no new
equilibrium condition can be reached.
Fig. 3. Glowing gaseous streamers
of an extinct titanic supernova
explosion of a massive star in
Cassiopeia A (Cas A) (observed by
the Hubble space telescope).
But even before that state is reached, it appears that the process becomes unwieldy and gigantic
explosions can take place. In the case of a main sequence star, like our sun, the first transition process
from hydrogen to helium fusion is accompanied by formidable explosions that eject the outer
envelopes of the star. Remnants from past explosions of this kinds are visible in the sky as planetary
nebulae. The most extreme kind of such an explosion is a super nova in which the force of the new
nuclear fusion reaction is so powerful that a large part of the star is blown away. During the few days
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of that explosion a supernova can emit more light than the entire galaxy to which it belongs. This
means that the absolute magnitude of a super nova is 1011 higher than our sun. Super novae are
therefore clearly visible. Far distant galaxies suddenly become visible during a super nova, after which
they again fade away to oblivion.
A well-known historical example of a super nova is the Crab-nebula, see figure 4. It was registered in
1054 by Chinese astronomers. During the super nova this phenomenon was so bright that it was
visible to the naked eye during day-time.
Fig. 4. Composite image of the Crab
Nebula showing superimposed images of
X-ray (blue) (by Chandra X-ray space
telescope), and optical (red) (by the
Hubble space telescope).
3.5. Remnants of Stellar Evolution
The location of the Crab super nova fom 1054 is nowadays identified as the Crab nebula, see figure 4.
The stellar remnant can also be identified as a faint star central in the nebulae. The Carb nebula
represents the ejected outer envelopes of the former star, and in fact they rapidly expand through space
as becomes visible in infrared light using the Doppler-effect.
Assignment 15: Design an empirical method using local observations of the Crab nebula in the
electromagnetic spectrum that would demonstrate that it is indeed expanding, and moreover
provide a method for estimating the expansion velocity from these method.
In the past decades observations with radio telescopes have shown that this central component emits
strong electromagnetic pulses with an extreme regularity6 of 33 ms, see figure 5. For this reason such
astronomical objects are called pulsars. The mechanism of these pulses is based on the search light
principle. The stellar remnant is spinning with great velocity.
6
As this phenomenon was discovered in 1967, in the first instances the discoverers thought it was a sign of
extraterrestrial intelligence.
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Fig. 5. First published registration of a pulsar, Hewish et al., Nature 217, p. 710, 1968.
Moreover it has a strong magnetic field that continually captures debris. The debris is guided by the
magnetic poles, where it emits strong radiation as it is accelerated in its fall to the surface. This creates
two strongly focused diametrically opposed beams radiating outward from the poles. However, the
magnetic axis and the rotational axis of the pulsar do not coincide. For this reason, the beam rotates
around the rotation axis. If we are inside a beam we are able to detect the radiation – if we are outside
we can not. This generates the pulses of radiation that we detect.
Assignment 16: Argue under which conditions we would observe a double frequency of EM
pulses as compared with the rotation frequency.
The Crab pulsar is a clear example of the final products of stellar evolution. There are several types of
stellar remnants, and they predominantly depend on the mass of the original star. For main sequence
stars as the sun, life ends with a super nova. In this event much of the total mass of the star is ejected.
The remaining mass contracts and can reach a stable state called a white dwarf. The stable state is
reached by a quantum mechanic effect called the Pauli principle. It results in a pressure generated by
electrons that can not occupy the same quantum state – the Pauli pressure. The pulsars, mentioned
before, all are white dwarfs. This is the normal final stage for most main sequence stars. It will be
reached if the remnant after the super nova has a mass § 1.3 Mü. For even smaller masses,
electromagnetic forces, like the van der Waals-force, can resist gravity, and the object will become a
brown dwarf or a planet.
Assignment 17: What is the mechanism that stops planets such as earth from imploding?
Massive stars like blue giants have a large energy output and short lifetimes of several million years.
Because of their huge masses their explosive potential is much greater than from main sequence stars.
13
Nevertheless, their end products are also more massive. Above the limit of 1.3 Mü there is nothing that
can halt the implosion of the star – at least at present their is no known physical law that could stop the
collapse. Thus the collapse continuous and the star becomes infinitesimal small and infinitely
compact.
Assignment 18: The escape velocity from a body B is the velocity vesc an object needs to have
in order to reach infinity, when launched from the surface of B. It can be found from an energy
2
consideration. The kinetic energy of the object at the surface of B is: E kin = 1 mv esc
2
and when reached infinity the kinetic energy is zero: E kin = 0 . According to Newton’s law of
gravitation, the gravitational potential energy of the object at the surface of B is:
E pot = −GmM / R , and when reached infinity it is zero: E pot = 0 . From the conservation of
energy find an expression the escape velocity vesc . At what radius R will the escape velocity
have reached the light velocity c? Suppose the object has one solar mass. Express the radius at
which the escape velocity becomes c in these variables. This radius is called the Schwarzschildradius. Calculate the Schwarzschild-for an object of one solar mass, and also for an object of
your own body-weight, use G = 6.7 10-8 cm3g-1sec-2.
From assignment 18 we find the so-called Schwarzschild-radius, i.e. the radius where the escape
velocity becomes the velocity of light c = 3.108 m/s. As soon as the object has collapsed within this
radius, even light can not escape from it. Hence, such an object is called a black hole. As we know
from Special Relativity, no material object can reach or surpass the velocity of light. Therefore,
everything falling to a black hole beyond the Schwarzschild radius is doomed. Entering the realm of a
black hole requires knowledge of both General Relativity and Quantum Physics. However, both
theories contradict each other at these scales – therefore there is at present no theory that can
adequately describe the interior of a black hole.
3.6. Stellar Evolution and the Hertzsprung-Russel Diagram
The HRD is very convenient for comprehending stellar evolution. From computational models and
observations we find that during its main stable state, the hydrogen fusion, a main sequence star
travels alongside the main sequence in the direction of the upper left corner, see figure 6. This passage
continues until about 10% of the amount of H has been transformed to He. At that time it travels
horizontally to the right (point A in the HRD), and then via B and C to the upper-right corner where it
enters the realm of the red giants (area D in the HRD). In the subsequent stages of nuclear fusion it
moves horizontally to the left (via E and F) until it almost again reaches the main sequence, and then it
explodes in a super nova (point G), leaving a remnant and a planetary nebula (area H in the HRD). If
the remnant is a white dwarf, such as a pulsar, its luminosity and temperature will give it a
characteristic place in the lower-left corner of the HRD (area J).
The age where a star with mass M and luminosity L leaves the main sequence in point A is
approximately 2.1010 M/L.
14
log L/ Lü
log Teff in K
Fig. 6 Path of the stellar evolution of a main sequence star of one solar mass in the
Hertzsprung-Russell diagram
Otherwise, we can also empirically validate these computational models by observing a cluster of
stars. All stars in a cluster have about the same age – the age the cluster formed – the same
composition (in terms of He/H ratio), and the same distance to the sun. Therefore, a HRD of a cluster
of stars can be made straightforward, see figure 7.
The main difference between stars in a cluster is based on their mass. Therefore we see a scattering of
stars over the main components of the HRD described above. Especially the main sequence is clearly
visible as stars of all masses are depicted in their travel up-left on the main sequence. The results from
these observations agree with the theoretical predictions and provide an upper limit for our sun of
approximately 1010 years (i.e. point A in the HRD). These results are shown in figure 7.
15
L/ Lü
sun
surface temperature (K)
Fig. 7 The HRD for 10 stellar clusters. At right ordinate the age in
billion years of the bifurcation point from the main sequence.
Fig. 8 Abundances of chemical elements in the
neighbourhood of our sun. The marks are from the
intensities from spectral absorption lines in the sun’s
atmosphere, the lines from meteorite and terrestrial
data.
16
3.8. Unstable Stars
From the onset of core Helium burning, stars move along the main sequence in the HRD. At
the end of their lives, stars proceed from the main sequence towards the area of the red giants.
During this transition, massive stars end heir existence in one single event, a super novae.
Fig. 9. An example of an unstable
– but not-periodic – star is this
massive ‘Wolf-Rayet star’
NGC2359, that irregularly ejects
large parts of its own outer
envelope in gargantuan
explosions. The star itself is in the
central bubble, the clouds are
remnants of previous ejections.
Low-mass stars, on the other hand, may transform less violently into red giants. However,
they can become unstable. This can express itself by huge explosions which we observe as
brightness fluctuations. These fluctuations can be erratic or periodic. A periodically
fluctuating star is called a pulsating star. In the HRD there is a specific region in-between the
upper main sequence and the red-giant group that is called the instability strip. When an aging
star passes through the instability strip its luminosity starts to pulsate periodically.
instability
strip
Cepheids
RR Lyrae
main
sequence
long period
variables
Luminosity
surface temperature
Fig. 10. Variable stars in the HRD. Pulsating variable stars
are found in the instability strip connecting the main
sequence and the red-giant region.
17
An example of a pulsating star is the Cepheid variable star7. A Cepheid star pulsates because
its outer envelope cyclically expands and contracts with a well fixed period.
Assignment 19: Argue how you can employ the Doppler effect and spectral lines in the
spectrum of a Cepheid to validate this assumption.
Moreover, Cepheid variables have a two important characteristics. First, they are very
luminous, ranging from 102 to 104 Lü. This makes that they are visible from large distances.
Secondly, they exhibit a clear relation between their period and their absolute luminosity.
Assignment 20: Argue how you can utilize the period-luminosity relation of Cepheids for
estimating their distance.
Cepheid Luminosity-Period Law
5
log(L/Lsun)
4.5
Fig. 11. Relation between luminosity and
oscillation period for Cepheid type 1
variable stars.
4
3.5
3
2.5
0
20
40
60
Period [days]
80
100
7
Named after its prototype, the star d Cepheid, discovered in 1784 by the then 19-year old deaf and mute
English astronomer John Goodricke, who died on the eve for his twenty-second birthday due to a pneumonia
contracted during his nightly observations.
18
4. Gravitational Fields and Stellar
Dynamics
All movement in space is governed only by gravitational interaction. This is on its own quite
remarkable, because of the three fundamental interactions known to us, the force of gravity is by far
the weakest. The strongest force we know of, is the force that holds together the atomic nucleus. For
this reason, it is called the strong interaction. The electro-weak interaction is responsible for the
electro-magnetic forces and the so-called weak interaction, responsible for e.g. the beta-decay. If we
compare the relative strength of the strong, electromagnetic, weak and gravitation interaction we find
about: 1 : 10-2 : 10-5 : 10-38. We see that gravitation is considerably weaker than any of the others, so
much that it appears that it could be neglected. In fact, however, the strong and electro-weak
interaction appear to be relevant only on small scales. On astronomical scales, therefore, only this very
weak force is relevant. The relative weakness of the gravitation causes that its effect only become
considerable when large amounts of mass are involved. This is visible in table 1, which lists the
masses of the planets.
Assignment 1: Both the forces of gravitation and electrostatics between two bodies
separated by a distance r, decrease with r as: r –2. This means that electromagnetism
36
remains 10 stronger than gravitation, irrespective of the distance two bodies are
separated. As both the and earth contain many charged particles, notably electrons
(respectively 1033 and 1031), why it is that the motion of the moon relative to the earth is only
governed by the law of gravitation?
Let us first consider the empirical laws of planetary motion stemming from detailed astronomical
observations. Next, we will examine the law of gravitation. Then, combining the laws of motion and
law of gravity, we will study its effect on motion in the universe.
4.1. The Laws of Kepler
Ever since man looked up to the sky and discovered the astounding exact regularities of celestial and
planetary motion, he wondered about the underlying laws and principles. For the Greeks, as for most
ancient cultures, the flat earth ruled at the center of the rotating universe. The, planets – from the
Greek word for ‘wanderers’ – though, posed a bit of a problem. Their irregular motion in the sky could
only be understood by invoking the epicycloid mechanism, that made planets move according to a
doubly combined rotation: a rotation according to an epicycle which center moved around the earth in
an orbit called the deferent, see figure 1.
Fig. 1 Epicycle model of planetary motion relative to the earth.
19
In the third century B.C. the Greek philosopher Aristarchos proposed a simpler – hence more elegant –
solution by proposing the sun as the center of celestial motion. In the middle ages this theory became
lost, but it was rediscovered by the Polish monk Nicolaus Copernicus (1473-1543), as the Heliocentric
model8.
Assignment 2: Can you explain the observed epicycloid motion of planets in the heliocentric
model?
Copernicus’ model motivated the German astronomer Johannes Kepler (1571-1630) to look for the
mathematical laws which governed planetary motion. His approach to the problem was essential
modern, and he belonged to the first modern scientists in that he strived to: (i) construct the best
(mathematical) model that could account for all the essential facts discovered in (ii) observationally
obtained data. In the possible multitude of models he chose the one that obeys Occam’s razor: the
most simple one9. As empirical data he obtained the best observations available at that date, those of
the Danish astronomer Tyho Brahe (1546-1601)10. Kepler was able to formulate the underlying
principles in three laws, that ever since bear his name:
Lex I: The planets describe elliptical orbits, with the sun at one focus.
Lex II: The position vector of any planet relative to the sun sweeps out equal areas of its
ellipse in equal times.
Lex III: The squares of the periods of revolution are proportional to the cubes of the
average distance of the planets to the sun.
These laws describe planetary motion with the greatest possible precision of his day, and allowed
accurate predictions of their positions.
4.2. Newton’s Law of Universal Gravitation
Now that the empirical facts of celestial motion were known in the phenomenological laws of Kepler,
the next step in the history of astronomy was to find an underlying mechanism that could explain them
in terms of a few basic principles. Here is where Sir Isaac Newton (1642-1727) made his outstanding
contribution, the law of universal gravitation. Second to his formulation of the physical laws of
dynamics, this discovery was his greatest contribution to the development of physics. It appeared as a
chapter in his monumental work Philosophiae Naturalis Principia Mathematica in 1687 – short the
Principia.
His starting point was his principle of dynamics: motion of a particle is caused by a force acting on
that particle. This force, F, changes the momentum p = mv in the period dt that it acts on it. Moreover,
let us also consider the directions of the force: F, and the velocity of the particle v. Let dp represent
the change of the momentum p, then Newton’s law of dynamics states:
dp
= F(x )
dt
(4.1)
Here, p = mv, and F varies in space depending on the position vector x.
In short, Newton’s line of reasoning for the law of universal gravitation was:
1. the force associated with gravitational action is central, i.e. it acts along the line joining the
two interacting bodies.
8
Copernicus was wise enough to let his work be published but after his decease, in order to avoid problems with
the clerical authorities.
9
Or, paraphrasing Albert Einstein: ‘A mathematical model must be as simple as possible, but not too simple.’
10
Tyho Brahe was rather reluctant to hand over his data, because he feared that all credits for finding the general
physical principles it contained would be earned by Kepler. Unfortunately for him, history proved him right.
20
2. The gravitational interaction is a universal property of all matter.
Because of his second point, Newton supposed that the gravitation force F was proportional to the
amounts of matter of the bodies; i.e. their masses m1 and m2. Newton’s universal law of gravitation can
be stated as:
The gravitational interaction between two bodies can be expressed by an attractive
central force proportional to the masses of the bodies and inversely proportional to the
square of the distance between them.
Or as mathematical expression:
Fgravity (r ) = −G
mM
r2
uˆ r
(4.2)
Where F is the vector describing the gravitational force that an object of mass M in the center of a
coordinate system, exerts on an object with mass m at position r in the coordinate system. Here r
represents the length of position vector r, and ur a unit vector – i.e. a vector of length 1 directed along
vector r. Note that ur can be written as: ur = r/r for r ∫ 0. The proportionality between the force and
the right-hand side is expressed in the constant G which in SI-units is:
G = 6.67.10-11 N m2/kg2
The fact that this constant is so small expresses the weakness of the gravitational interaction. G is a
fundamental constant of nature, just like the velocity of light c = 3.108 m/s, the proton charge e =
1.6.10-19 C, the rest mass of the electron me = 9.1.10-31 kg, and the constant of Planck h = 6.6.10-34 J s.
As yet, there is no known underlying mechanism to explain why these constant happen to have just
these values, but if they would vary as much as 10-9 the resulting strengths of their interactions would
not yield stable atoms, no molecules, no life, and hence no intelligent life as we know it to observe it.
The universe would be filled with radiation and uncoupled elementary particles.
Assignment 3: Estimate the mass of the earth from the law of gravitation, using that earth has a
radius of 6.37.106 m, and the acceleration of gravity at the earth surface is: 9.8 m/s2.
4.3 Gravitational Potential Energy
Since the gravitational interaction defined by equation 4.2 is central and depends only on the distance,
we may associate it with a gravitational potential energy. This is similar to the electrical potential
energy. Interactions with these characteristics are called conservative. For conservative interactions,
the interaction force may be written as the negative gradient of the interaction potential energy Epot.
Therefore, we may write:
∂E pot
∂r
= − Fgravity (r ) = G
mM
r2
uˆ r
(4.3)
The solution of this equation yields:
E pot = −G
mM
r
(4.4)
Here we assume the potential energy to be zero at for infinite separation.
21
4.4 Dynamics Resulting from Gravitational Interaction
We can now study the motion of N isolated particles due to gravitational interaction. The total energy
of a such a system is:
E = ∑ particles ½mi v i2 − ∑ pairs G
mi m j
rij
(4.5)
Such a system may model the motion of the solar systems with the sun, the planets and the comets. Let
us now study a system containing two particles in more detail. Such a assemblage is called a binary
system. An example we bear in mind is the sun-earth system – temporary ignoring all other members
of the solar system. Let us assume that one mass is much larger than the other; M à m. We may than
approximate the energy as:
E = ½mv 2 − G
mM
r
(4.6)
Here, r and v are respectively the position and the velocity of small mass m relative to the large mass
M. In expression the term E is a constant, because of the conservation of energy. Therefore, there are
three possibilities for a binary system.
Fig. 2 Possible trajectories in a gravitational field for different values of the total energy.
1. E < 0. This represents a bound system. The bound nature of the dynamics means that the
kinetic energy at any point of the orbit is insufficient to take the small mass to infinity. This
generally results in a elliptical path of the small body around the larger mass.
2. E > 0. This represents a free system. The kinetic energy is sufficient to bring the small mass to
infinity, and after some time it will travel with a uniform velocity. This situation results in a
hyperbolic path of the smaller body.
3. E =0. This represents the boundary case between the former two extremes. The kinetic energy
is neither sufficient to entirely free the body from the gravitational field, nor will it ever
complete a revolution. In practice, this situation will never be reached because the probability
to set v to the required value is zero. The resulting trajectory is a parabola.
22
4.5 The Gravitational Field
An important concept in physics is the notion of a field. We can assign a field called the gravitational
field to the gravitational interaction. The gravitational field strength G produced by a mass M at point
P with position r is defined as the force exerted on a unit of mass placed at P. Thus the gravitational
field G always points towards the mass producing it. The force F a body of mass m experiences in a
gravitational field G therefore is: F = mG. Associated to the field is a gravitational potential f, such
that the potential energy Epot of a mass m in the field equals: Epot = mf. Because of equations 4.3 and
4.4 we may write:
G=−
M
∂φ
= −G
uˆ r
r
∂r
(4.7)
The concept of the gravitational field enables us to introduce two important characteristics of
gravitational fields, see figure 10. Libration points are the three optima in gravitational potential,
here denoted as L1, L2, and L3. In the central libration point, L1, the field vector G is zero. The
Roche surface is the horizontal 8-shaped surface that envelopes the two masses. Within the
Roche surface, small masses will fall to the mass to which the segment of the field belongs. If
a star expends, e.g. in the red giant phase, and traverses the Roche surface, its mass starts
flowing to the other component. This mass overflow will result in the release of potential
energy which generally escapes as violent bursts of X-ray radiation, which are clearly visible
in the sky – given suitable equipment.
Fig. 3. Libration points and
Roche surface in the
gravitational field lines of the
masses.
4.6 Orbital Motion in our Solar System: Planets, Comets, and
Satellites
Consider a collection of rotating and moving bodies. Let L denote the angular momentum of a body,
and h its angular inertia. Then the total gravitational, kinetic, and rotational energy of the collection is:
E = ∑ particles
mi m j
pi2
L2
+ ∑ particles i − ∑ pairs G
(4.10)
rij
2mi
2η i
All celestial motion can now be understood as the result of dynamic motion caused by inertia,
rotation and the gravitational interaction – defined in this equation. These laws have been very
successful in determining complex dynamical motions caused by gravitational fields.
Examples of such applications are:
Satellites Trajectories
Using equations like (4.10) we can exactly plan an interplanetary flight with great
precision. The mathematical tools used are all provided by Newton in 1687, and have
not been changed since!
23
Binary Star Systems
In the case of two stars revolving around their center of mass, we can use the equation
of motion to obtain useful expressions for the total mass and the radius of the system.
Galactic Disks
Orbits of individual stars in galactic disks obeys Newtonian laws of gravitation and
kinematics.
Large Scale Movement in the Universe
Large scale movements of individual galaxies and clusters of galaxies follow
Newtonian laws just like they were pointsources.
Assignment 4: Confirm the validity of the three laws of Keppler for a perfect circular and
uniform motion of a body with mass m orbiting a central body with mass M. Use the laws of
kinematics and the expressions for gravitational and centripetal force. Consider M à m, such
that we can consider the center of the motion fixed in the center of the large body.
Assignment 5: Consider a perfect homogeneous spherical black body of large mass M. Suppose
that the body spins with small revolution time. Is there an experimental way of finding out
whether the body rotates.
Assignment 6: Suppose our sun collapses to a neutron star with a radius of 10 km. Calculate
the new rotation time, starting from the present sidereal rotation time of 31 days.
24
5. Exotic Matter in the Universe
In recent years it has become clear that most matter in space is not in the form as we know it; as
ordinary matter or as ionized matter in stars and interstellar clouds. In fact, all matter we can observe
and detect in the universe can only account for some 10% of the total mass that must be available in
the universe. The missing mass is called Dark Matter. Several proposals have been made to explain
the conundrum of dark matter. One explanation regards bodies that were too light to form stars, but
much more heavy than planets. Such bodies are called Brown Dwarfs, and they are near-undetectable.
Another form of difficult to detect matter is in the form of old pulsars. These are massive, but faded
away and there rotation has almost stopped, which makes it difficult to detect them. Other more exotic
possibilities are also considered. Black holes occur when no force can resist gravity. By their very
nature they are dark, for not even light can escape from its inner sphere – hence its name. Dark matter
was first identified in the halo – the sphere surrounding its kernel – of galaxies. One suggestion for
dark matter in these halo’s are Massive Compact Halo Objects, short MACHO’s. Indeed, MACHO’s
have been detected. As they themselves are dark, the only way to detect them is because their strong
gravitational fields bend light, and so they diffract the light of stars that are positioned behind them.
This effect is called gravitational lensing. This phenomena has indeed been observed in the halo of
our galaxy, and are a good indication of MACHO’s.
A final suggestion concerns fundamental particles that are relatively massive, but do hardly interact
with ordinary matter. Such particles are called WIMPS: Weakly Interacting Massive Particles. A
similar example is the neutrino, it is not so massive, but recent experiments suggest that it carries a
very minute amount of mass. Since there are so many neutrinos in the universe, the total amount of
mass in the neutrinos is considerable – but still insufficient to account for all dark matter. perhaps our
understanding of physical laws is not as complete as we think.
5.1 Detection of Dark Matter in Galaxies and Clusters
As we saw, the laws of Keppler can adequately describe the motion of planets in their orbits around a
star. Similarly, the laws of universal gravitation describe the motion in a galactic disk. A star moving
in a galactic disk is totally determined by the gravitational pull of all other stars in the galaxy.
Consider a star in a circular motion in a galactic disk as in figure 1 below. The orbit of this star is
given as the dotted line. Some of the gravitational pull on the star by the other stars in the disk is
shown in the figure 1: nearby matter pulls strongly, matter far away is more numerous, but because of
the larger distance and the 1/r2-law the pull is much weaker. Now one can demonstrate that the
gravitational pull of all matter of the galaxy outside the orbit (indicated as gray in the figure) cancels
exactly. therefore, the gravitational pull is determined solely by the mass inside the orbit of the star.
in
Fig. 1. Orbit of a star in a galactic
disk, and gravitational forces from
objects outside the orbit acting on
the star.
out
For this reason, the period of the star is an indication of the mass inside the orbit.
Assignment 1: How could you measure the period (revolution time) of a visible star in the
galactic disk?
25
The curve that shows the orbital speeds of stars and gas in the disk of a galaxy versus the distance to
the galactic center is called the Galaxy Rotation Curve. Using this curve and the known laws of
gravitation and kinematics, we can calculate the matter inside a given radius of the disk. However, the
calculated mass, required mass to explain the orbital motion, is ten times higher than the mass that is
actually observed! The missing matter is called dark matter because we can perceive its existence only
through its gravitational influence on the stellar orbits in the galactic disk.
The same situation occurs in galactic clusters and superclusters. Similarly, we can estimate the visible
mass of the constituents. Alternatively, we can infer their masses also by using the laws of gravity and
kinematics. Again, we find that the required mass for the observed dynamical orbits is ten times as
high as the actually observed matter. Again, 90% of the matter is dark matter.
Finally, as we will later see, the entire universe is expanding. From observations of the dynamics of
this expansion and using a model for gravitational interaction11 we can calculate that even much more
of the required mass is missing.
Summarizing we conclude that most matter in the universe is in the form of dark matter. All proposed
explanations, WIMPS, MACHO’s, black holes, extinct pulsars, brown dwarfs, neutrino mass, can only
contribute to a small part of the required mass. Perhaps our basic description of nature must be
revised.
5.2 Supermassive Black Holes in Galactic Centers
In the center of our own Milky Way the density of stars is hundreds of times higher as in our own
neighborhood, which is in the outer rim of the galactic disk. Based on the observed motions of stars,
the galactic nucleus is situated in the constellation Sagittarius. It has been known for some time now
that one of the most powerful radio sources in the sky is located at this location. This source is called
Sagittarius A. Due to intergalactic dust clouds, it was until recently impossible to directly observe the
galactic center. Nowadays, using infrared light and radiowaves, we can make good images of the
galactic nucleus. These observations show that Sagittarius A is composed of multiple sources, from
which the strongest one is thought to be the galactic nucleus. This source is called Sagittarius A*. The
inner sphere of Sagittarius A spans about 20 lightyears across and contains several thousands stars.
Recent observations show fast motions of the stars very close to Sagittarius A*. These observations
show that these stars have speeds of more than 1500 km/s. Obviously, there must be a very massive
body that binds these stars in orbits. Using Keppler’s third law and Newtonian dynamics, it is possible
to estimate the mass of the central body. These calculations give a mass of approximately 3.106 Mü .
Yet detailed observations of radio source A* show that this mass must be concentrated in a volume
less than our solar system. therefore, it seems logical that this mass can only be a supermassive black
hole.
Observations of other galaxies, especially of active galaxies like Quasars indicate that most galaxies
have supermassive black holes in their nucleus. In active galaxies these nuclei devour large numbers
of stars. As these stars fall into the black hole, they emit large amounts of radiation. Even the nucleus
of our galaxy regularly consumes a star, thereby releasing huge quantities of radiation. This also is the
reason why Sagittarius A is such a strong radio source.
Though these black holes are enormous massive, their masses by now means can compensate for the
missing dark matter.
11
Here, the gravitation is not described by Newtonian gravitation but by a geometric theory called General
Relativity, introduced by Albert Einstein.
26
6. The History of the Universe
In chapter 1 we saw how the known universe is hierarchically built upwards from meteorite- and
planetary-sized objects up to large-scale structures stretching for hundreds of millions of light-years.
We will now consider how astronomy currently understands the structure and formation of the
universe. Is the universe infinitely large and infinitely old? Or is it finite in time and space?
6.1 The Infinite Static Universe
Let us first consider the question whether the universe is infinitely large. However, we first have to
specify our conception of universe. In colloquial language ‘universe’ both relates to the fabric of space
and time, as well as to the distribution of physical substance (matter and energy) in space. It is
conceivable, for instance, that only a part of all space is actually filled with interesting stuff like matter
and energy, and the remainder absolutely empty. However, both extremes – a totally filled space and a
partially filled universe – lead to paradoxes. Let us, therefore, make a distinction between spacetime
(as we have learned from relativity theory) and the substance filling the spacetime. Regarding the
space encompassing the universe we propose the so-called the cosmological principle: i.e. we assume
that all fundamental characteristics of space are isotropic and homogeneous. The substance filling
space, however, is inhomogeneously distributed as discrete clumps of matter (planets, stars, galaxies)
with wide voids of empty space.
Now, suppose that the substance filling the universe stretches out infinitely far in about the same way
as the visible universe. In that case, at large scales, if the universe continues in the same way as in our
vicinity, the universe would become uniformly distributed. The planets, the stars, and even the
galaxies would become but minor impurities in the otherwise homogeneous universe. Thus, in every
possible direction that you would look, sooner or rather later, there would be some luminous object.
So, from every possible direction light would meet our eyes. Therefore, the entire night sky would be
as bright as the surface of the sun. Clearly, it is not. This circumstance is called Olbers’s paradox12.
Obviously, our starting point was incorrect. Either the universe does not stretch out infinitely, or at
some distance the density of luminous objects significantly decreases from our local one.
On the other hand Newton came with yet another – seemingly – persuasive argument for an infinitely
large and static universe. As we saw in earlier chapters, on large scales the universe is dominated by
the attractive force of universal gravitation. Therefore, all matter would fall together into one big
clutter and the universe would contract to an infinitely small size. How then do not all celestial objects
fall towards each other – or rather – have not cluttered already? This predicament was of great concern
to Isaac Newton, the very inventor of both the laws of dynamical motion as of the laws of universal
gravity. As every man of his age, since the days of Ptolemy13, he was strongly convinced of a static –
in the sense of unchanging – universe. To resolve this dilemma he argued that in an infinite, uniformly
distributed universe, the gravitational force on a star would act from all possible directions with equal
strength, and therefore would cancel exactly. This indeed would make the universe static, but as a
direct consequence it would have to be infinite and homogeneous. However, this would again lead to
Olbers’s paradox!
Assignment 1 small perturbations in a static universe: How would such a static universe react
to small and local perturbations in the distributed mass? What do you then conclude about the
viability of this model for a static universe.
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After the 19th century German astronomer Heinrich Olbers
The last of the great Greek Astronomers who lived during the second century A.D., and constructed a model of
the universe where the earth was set at the center of the universe, and all other bodies (moon, sun, planets, and
stars) where fixed on rotating concentric celestial spheres.
13
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From these arguments alone, a stable static universe seems infeasible. Let us now see what
observations in the past century have taught us.
6.2 Hubble’s Law of Redshift
Early on in the 20th century, scientists argued whether the universe is uniformly filled with stars and
whether galaxies are just some kind of nebulae (e.g. dust clouds), or alternatively whether galaxies are
colossal collections of stars, and our sun, together with the visible stars, constitute an equivalent
aggregation; our own galaxy the Milky Way14. For this reason much attention was devoted to the
observation, analysis, and modeling of galaxies. Two American astronomers, Edwin Hubble and Vesto
Slipher made a series of important discoveries. First, by 1920 Slipher had discovered that the
overwhelming majority of the galaxies that he observed exhibited spectral lines that are shifted
towards the red end of the spectrum. Employing the Doppler-effect this means that most of the
galaxies are receding from us. Second, in 1923 Hubble in analyzing a series of photographs of the
Andromeda Nebula – the closest galaxy to our own – discovered some distinct Cepheid variable stars.
As we saw in chapter 3, Cepheid variables are luminous pulsating stars that exhibit a consistent
relation between the period and absolute luminosity. Using the Cepheids as standard candles, i.e. as a
gauge for establishing distances, Hubble gave the first decisive proof that galaxies are indeed much
more distant than the visible stars, and that they themselves consists of enormous numbers of stars.
Consequently it became clear that our Milky Way is also a galaxy. With the Cepheids as yardstick he
could now confidently measure the distance to nearby galaxies, namely the galaxies exhibiting
Cepheids.
Assignment 2 estimating the distance of an observed Cepheid: Using the Hubble Space
Telescope, a team of astronomers in 1992 found a Cepheid variable in a galaxy named IC4182.
This Cepheid had a period of 42.0 days and an average apparent magnitude of m = +22.0. From
this figures and the Cepheid period-luminosity relation as depicted in figure 11 from chapter 3
estimate the distance from this star – and so its galaxy – to earth.
For a number of galaxies so close that they allowed the detection of individual pulsating stars, using
the period-luminosity relations for pulsating stars, he determined their distance to earth.. Now, using
the observation of Slipher, Hubble plotted the recessional velocity of these galaxies – calculated from
their redshifts using the Doppler-effect – against their distance to earth. What he found was a
revelation; there appeared to be a very distinct linear relation between their recessional velocity and
their distance to earth. This relation is since known as Hubble’s Law.
We can formulate Hubble’s law as follows: two galaxies separated at a distance d recede from one
another with a velocity v that obeys:
v = H0 d
In this formulation H0 is a constant called Hubble’s constant. Based on currently available information
the value of this constant is:
H0 = 70 km/s/Mpc
So, two galaxies separated 1 Mpc recede with a velocity of 70 km/s.
Assignment 3 receding velocity of IC4182: Using the distance you found for galaxy IC4182.
above in assignment 2, calculate the velocity it moves away from earth using Hubble’s law.
14
The discussions whether galaxies were mere nebulae or distant and colossal star systems found its culmination
in the ‘Shapley-Curtis debate’ in the 1920s.
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Hubble's Law of Redshifts for 36 galaxies
350
velocity in km/s
300
250
200
150
100
50
0
-50
distance in Mpc
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
Fig. 1. Hubble’s law of redshifts for 36 galaxies. The redshift is calculated to the corresponding
receding velocity using the Doppler effect.
6.3 The Expanding Universe
What does the law of Hubble teach us about the nature of the universe? Superficially, we could
conclude that we have restored the heliocentric model; we (the sun) is at the very center of the
universe and all other galaxies are receding from us according to Hubble’s law.
Assignment 4 what the principle of Newtonian relativity teaches us about our place in the
universe: Suppose that all galaxies in the universe neatly obeyed Hubble’s law. In Newtonian
relativity all physical laws are equivalent on all positions in space and time, even if observers
were moving relative to each other with constant speed. Argue how Hubble’s law would be
formulated from the stance of a galaxy at one million lightyears distance from us?.
The last assignment shows us that there is no real center of the universe. From all galaxies in the
universe it would appear whether all other galaxies were receding from them.
Assignment 5 about the linear character of Hubble’s law: Suppose that the law of Hubble was
formulated as: ‘all other galaxies are receding from us with constant velocity irrespective of the
distance.’ Would such a law obey Newtonian relativity, i.e. would it be stated equivalently
disregarding your position in universe?
To interpret Hubble’s law, let us use a simplified analogy for the expanding universe. Suppose that
you observe an exploding cloud of shrapnel. Consider the individual bullets as galaxies, and observe
how the cloud expands in empty space. Now observe how the individual bullets recede from one
another. To avoid problems of interpretations at the
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Assignment 6 shrapnel analogy: Demonstrate that in this model the individual bullets follow
Hubble’s law.
All the bullets in the cloud recede from one another as the cloud expands, just as all galaxies recede
from one another as the universe expands.
This analogy shows us that:
1. all galaxies recede from one another with a velocity that increases with their distance
2. there is no center of the universe
Einstein’s General Theory Of Relativity
At the time as this information became available, the great physicist Albert Einstein had already
completed his general theory of relativity. In contrast to his special theory of relativity which
formulated physical laws in systems moving uniformly relative to each other, Einstein here described
the physics of relative acceleration and gravity. Einstein started from a simple observation, the
equivalence principle – stating that we can not distinguish between uniformly accelerated motion and
a uniform field of gravity. From this principle he formulated a theory in which gravity intrinsically
affects the curvature of space.
Fig. 2. Einstein while writing down the
major equations of General Relativity. He
regarded the moment that he finally
understood the fundamental principle of
this theory he stated as ‘ ... the most
delighted moment of my life’.
A direct consequence of his mathematical theory was that there would be no stable universe.
According to the general theory of relativity, a uniformly distributed universe gave a solution of a
steadily expanding universe. This observation was to the great dismay of Einstein as he, like all his
contemporaries, was convinced of a static universe. Therefore, he did what all mathematicians do
when their model does not match observation – or like in this case his preconception. He added a
mathematical term to his formula that made the solution static. Note that his original ideas was based
purely on physical observation – the equivalence principle – and that now he performed a
mathematical trick without any basis in physics, just to fit the outcome with his beliefs. He called this
supplementary mathematical term the ‘cosmological constant’, denoted L. As the Hubble law was
formulated indicating a continuously expanding universe, Einstein realized that he had missed the
opportunity to predict that the universe necessarily was expanding, and that, in his words, ‘the
introduction of the cosmological constant was the biggest blunder in my life’.
The Geometry of Spacetime
The Special Relativity Theory deals with inertial frames, frames of reference that move with
uniform speed relative to each other. In the General Relativity Theory (GRT) Einstein
considers general frames of reference, including noninertial. His starting point was that:
‘The laws of physics must be of such a nature that they apply to systems of reference in
any kind of motion’.
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Combined with the equivalence principle he could make the following associations:
gravity fl accelerated motions fl noninertial frames fl curved spacetime
Following this chain of reasoning, one direct consequence of the equivalence principle is that
spacetime in a gravitational field is curved. This curvature is intrinsic, i.e. a property of
spacetime itself, however, we can visualize the curvature of spacetime best with an analogy.
Consider a universe consisting of 2 spatial dimensions and time. Now consider a massive
body M at the center of the coordinate system of this universe. In the Newtonian model the
space can be represented by a flat plane, with M in the origin. In GRT, however, space is
curved. We can symbolize this curvature by representing the space as a curved surface. The
body M here acts as a depression in the surface.
Fig. 3. Model of a 2D
universe curved in a third
dimension by the action
of a massive body
positioned at the centre
of the dint .
The curvature of space has all kinds of effects like the bending of light near massive bodies
and deformations of spatial dimensions and slower running clocks; clocks in gravitational
fields run slower.
6.4 The Big Bang Theory
A logical consequence from the model of an ever-expanding universe is that, looking back, at one time
everything in the universe was crapped together in an infinitely small region of space. Therefore, there
must have been a beginning of time, when space was infinitely small and dense, and the universe
started to explode. This moment is called the ‘Big Bang’15.
Assignment 7 Last departure of Andromeda: The Andromeda nebula or M31 (see figure 2 of
chapter 2) is the nearest galaxy to the Milky Way. Its distance to earth is approximately 2
million lightyears. Use Hubble’s law to predict its receding velocity V. Using this velocity V,
estimate how long ago we departed from M31.
Assignment 8 Last departure of M101: The beautiful spiral galaxy M101 (see figure 4) is the
binary galaxy approximately 27 M lyrs away from earth. Like in the previous assignment, use
Hubble’s law to predict its receding velocity and estimate how long ago we separated.
In the last two assignments we found that both galaxies separated at the same time from our own
Milky Way. Using Hubble’s law, we can estimate the time ago that a galaxy at distance d Mpc
departed from our own. This time is: T = d/v = d/H0d = 1/H0 . Note that this time is independent from
the distance d. The value is the same for all galaxies.
Using the value of 70 km/s/Mpc we find:
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T = 1.4 10 years.
15
The term was coined by the eccentric British astronomer Fred Hoyle, who was skeptical towards this idea and
in 1947 commented that: ‘ ... certain American theories let us belief that the universe start in a Big Bang.’.
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Fig. 4. M101, the ‘Pinwheel
Galaxy’ in the constellation of
Ursa Major is a nearly face-on
galaxy with a bright nucleus and
clear spiral shape. It is located
about 27 million light years from
Earth with an estimated diameter
of over 170,000 light years. It is
one of the largest disk galaxies
known. M101 is a bright object
with a magnitude of 7.9, and easily
visible with binoculars or small
telescopes.
Thus, according to this simple calculation, the Big Bang occurred some 14 billion years ago.
The concept of the Big Bang as origin of the universe is an inevitable consequence of Hubble’s
observation of an expanding universe. At the moment of the Big Bang, the universe was a constricted
to an infinitely small space, and hence, infinitely dense. This location in spacetime is a mathematical
singularity, comparable to the center of a Black Hole. Due to this singularity, we can not satisfactorily
model the phenomenon mathematically. Therefore, concepts as ‘here’, ‘now’, ‘past’ and ‘future’ loose
their meaning. Using General Relativity and Quantum Mechanics, however, we can estimate the time
after the Big Bang that our physical laws became applicable. This is the so-called Planck-time:
tPlanck = 1.35 10-43 s
From the start of the Big Bang to the Planck time we lack the proper tools for modeling the universe.
After that brief interval we can model the evolution of the universe using the fundamental laws of
Physics. Using this laws we can make some predictions that we can test.
The Early Universe
One of the consequences of the physical models just after the Big Bang is that the early universe was
extremely hot and opaque; i.e. light was not free to move as it was consistently absorbed. It was so hot
that thermonuclear fusion could happen spontaneously everywhere in the universe. From the
conditions in the early universe the physicists Dicke and Peebles could actually account for the
observed abundance of heavy elements in the universe. The hot early universe must have been filled
with numerous high-energy short-wavelength photons. The properties of these photons are well
modeled by the Planck model for blackbody radiation. Due to the continual expansion of the universe
the universe cooled. We can compare this cooling with adiabatic cooling of a gas by expansion in a
cylinder.
Models for the Evolution of the Universe
Using the General Relativity Theory we can make again a 2D-analogy of the expanding universe. To
interpret Hubble’s law, we consider a two-dimensional and closed model of the universe. Now
consider the following analogy. Suppose that you have a deflated balloon on which you mark irregular
spots all around. Consider these spots as galaxies and the surface of the balloon as empty space. Now
inflate this balloon uniformly and observe how the spots recede from one another.
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Assignment 9 balloon analogy: Demonstrate that in this model the spots follow Hubble’s law.
All the spots on the balloon recede from one another as the balloon expands, just as all galaxies recede
from one another as the universe expands. This analogy shows us that:
1. all spots recede from one another with a velocity that increases with their distance
2. there is no center of the universe
3. rather than an explosion of matter in empty space, space itself is expanding
The Critical Density of the Universe
The evolution of the universe is solely determined by the amount of mass available in the universe,
and the total amount of kinetic energy present during the big bang. In that respect the universe
resembles the orbit of a bullet that is shot in the air that is bound by gravitational energy. With more
than enough kinetic energy, the escape velocity of 11 km/sec, the bullet is able to escape the gravity of
earth and swiftly fly away from earth. Below this value it will fall back to earth. At the exact critical
value of the escape velocity it will fly away but at ever slower pace, and reach zero-velocity at infinity.
The situation for the universe is similar. Here, however, the critical parameter is the mass density of
the universe. There is a critical density rcrit above which the universe will collapse together into a ‘Big
Crunch’. Above the critical density it will expend for ever. If the density of the universe exactly equals
the critical density, it will expend, but at ever lower rate, until at infinity, it will stop. Using
cosmological models based on GRT, rcrit can be calculated as:
rcrit = 0.2 10-27 kg/m3
The Cosmic Background Radiation
After about some 300,000 years, the cooling of the universe had progressed so far that, rather abruptly,
the entire universe became transparent. Thus, at once light could travel all the way through space. That
light could be described by blackbody radiation, with its peak according to Wiens’s law. Since that
moment, now 14 billion years ago, the entire universe has expanded, so we must use adiabatic
expansion to calculate the temperature of that heat-distribution by now. Correct computations
predicted a value of about 3 K. This radiation must now be detectable as a continuous background
radiation. Since it was emitted some 300,000 years ago in all directions, we must now receive it
uniformly from all directions. For this reason it is called the Cosmic Background Radiation or CBR.
We can regard the CBR as the afterglow of the Big Bang. This CBR is all around us. In fact, it is even
responsible for a few percent of the noise in mobile TV-sets. In the 1960-ies two engineers of Bell
Labs, Arno Penzias and Robert Wilson, detected some annoying noise in their new and unprecedented
large microwave antenna. As they tried to figure out where the origin of the noise was, they found to
their astonishment that it was evenly distributed over the sky. They had never heard of the Big Bang,
but after some research they found out of this theory, and the predictions of the CBR. The peak of the
observed background noise corresponded to a temperature 2.725 K , after using Wien’s law. That was
a triumph for the Big Bang theory.
Slight variations in the Cosmic Background Radiation
As the early universe became transparent it was not entirely uniformly distributed. If it was, no
galaxies would have formed, and we would not be here. Small variations in the moments after the Big
Bang have become literary inflated to large density variations. These variations would later grow to
the condensation kernels for future galaxies. At the moment of emission of the CBR these fluctuations
were extremely subtle. In the last decennium, however, detailed astronomical observations have led to
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the detection of these variations. Since end 2002 a detailed map is available of the variations of the
CBR, so a snapshot of the baby universe at the young age of 300,000 years. This map is of great
importance for finding out the geometrical shape of the universe.
Fig. 5. Subtle variations
in the CBR. scientists
using NASA's
Wilkinson Microwave
Anisotropy Probe
(WMAP) during a
sweeping 12-month
observation of the
entire sky
The isotropy problem and the Inflation Theory
The variations in the CBR as shown in figure 5 are much less than originally expected. They are as
subtle as 1 part in 10,000. This means that the CBR is extremely uniform from all directions. This
conundrum is called the isotropy problem. This again, means that the temperature of the universe must
have been extremely uniform. However, the universe must by an age of 300,000 years already been
enormous large. A second problem is that the proposed density of the universe is close to critical
density, the density that would make the universe ‘flat’. This condition is the flatness problem. To
resolve this problem scientists have proposed the theory of inflation. In this theory they define a short
period in which the universe expanded exponentially to about 1050 times its size during only 10-24 sec.
This inflationary epoch occurred only shortly after the Planck time. This theory satisfactorily explains
both problems. At an instant after the big bang the small variations in the universe were inflated to
extremely large size, mimicking an almost uniform distribution of the background radiation and
seemingly making the universe appear as totally flat.
Accelerating Universe and Anti-Gravity
Another problem is that recent observations of distant super novae indicate that the expansion of the
universe is accelerating. This means that the Hubble law is not linear, but that the receding velocity v
increases more than linearly with the distance d. The reason for this discovery is that good standard
candles became available in the form of a special type of supernovae. Remember the role of Cepheids
as standard candles for determining distances for nearby galaxies. Since super nova are more luminous
than entire galaxies, these events can be observed at great distances. Thus, it was possible to exactly
determine the distances to a number of distant galaxies that exhibited these types of super novae. The
results of these observations indicate that our present theories for the evolution of the universe are
inadequate.
6.5 The Fate of the Universe: Big Crunch or Big Sleep
On the long run all models for the evolution of the universe present unattractive scenario for the
distant future. In case of sufficient mass in the universe to stop expansion, i.e. if the density is below
the critical density rcrit, the universe will collapse into an event that is called the ‘Big Crunch’, the
opposite of a Big Bang.
If the density is equal or above that value, the universe will expand for ever. All galaxies will either
merge or continue to recede from each other. So, it becomes more and more difficult to observe other
galaxies. All stars will end their lives whether or not as super novae. Most matter will be used in the
process of star formation, and thus end up as stellar remnants. Colossal black holes will devour
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significant amount of matter. After the last matter has been used for star formation there are no more
luminous objects in space and it becomes pitch dark. All matter that escapes the black holes will decay
into protons, neutrons, electrons, and radiation. If elementary particles as quarks and leptons decay,
these will also transform into radiation. In the very long run, due to the quantum tunneling effect, even
the black holes themselves will evaporate. This means that after some 1036 years the whole universe is
filled only with EM radiation. Due to the constant expansion, the photons will become of ever longer
wavelengths and lower energies. Therefore the EM radiation becomes ever weaker and weaker. Thus
this prospect ends in a boring event-less universe where time has lost its meaning.
Acknowledgements
This research has made use of NASA's Astrophysics Data System
Further reading:
For those interested in more documentation we highly recommend:
W. J. Kaufmann, (2002) Universe, 6th edition with CD-Rom, W. H. Freeman, ISBN 07167
38236.
Moreover, numerous splendid websites are available on the web.
Here are only a few:
1. http://www.nasa.gov/
2. http://hubble.nasa.gov/
3. http://www.bbc.co.uk/science/space
4. http:///www.esa.org
5. http://www.astro.lsa.umich.edu/Course/Labs/pleiades/pl_intro.html
6. http://nrumiano.free.fr/Estars/sequence.html
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