introduction to astrophysics - maastricht university · 2013-06-10 · 1. introduction astronomy is...

0

Introduction IInnttrroodduuccttiioonn

to ttoo

Astrophysics AAssttrroopphhyyssiiccss

Ronald L Westra

Maastricht University

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 lsquoastronomyrsquo itself derives from the Greek aster meaning lsquostarrsquo and nomos meaning lsquolawrsquo 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 Einsteinrsquos 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 ndash as yet undecided ndash observatory andor 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

1

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 lsquotheoriesrsquo 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 ndash according to our present model of the universe 21 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 ie 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 earthrsquos continents and at 107 m we see the entire earth with a diameter of 12107 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 76108 m At 31011 m we have the diameter of the earthrsquos orbit around the sun The distance to the sun is often used as yardstick called 1 AU = Astronomical Unit = 151011 m From the sun we find the planets Mercury Venus Earth Mars the asteroid belt Jupiter Saturn Neptune Uranus and finally at 601012 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 601012 m 30108 ms ~ 20000 seconds ie 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 ndash the distance light travels in 1 year = 3107 s micro 3108 ms = 9461015 m Astronomers prefer the so-called parsec (pc) = 326 Lightyears This is the distance of a star that virtually moves 1rdquo 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 = 326 lightyears

2

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 ~ 81019 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 ndash 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 fuzzygalaxies in the lower left of the figure are in theprocess of merging resulting in huge veils ofstars accompanying them The lsquosmallrsquo crispgalaxy almost in the center is actually on thebackground and is far more distant than thiscollection

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 lsquoripsrsquo the holes in continuous space In this sponge-structure is a remarkable formation called lsquothe great wallrsquo or the lsquocentral attractorrsquo It is the largest structure in the known universe and exerts its gravitational pull on all visible matter

3

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 ndash 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 lsquoSwiss-cheesersquo 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 micro velocity-of-light ~ 14109 years micro 3107 secondyear micro 3108 ms = 1261026 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 22 A Travel in Time

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 lsquoBig Bangrsquo If we follow the big bang theory ndash 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

Fig 5 The Universe at the young age of 300000 years The colors represent temperature fluctuations in the Cosmic Background Radiation (courtesy Wilkinson Microwave Anisotropy Probe)

1 only in first order it is linear

4

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 lsquocoupledrsquo Only after 300000 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 colossal super novae After about 10 billion years ndash 46 billion years ago ndash our solar system formed The formation of earth-moon system happened as the result of a primordial collision some 45 billion years ago A proto-planet about the size of Mars collided at high speed with the nearly fully formed Earth The collision shattered Earth and pulverized the 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 Saturns 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

Fig 6 The Giant Impact Theory suggests that a Mars-sized object crashed into the early Earth Most of the debris thrown into space fell back on Earth but a fraction aggregated into the Moon This theory is supported by the similar composition of rocks on the Earth and Moon (courtesy BBC))

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 ndash and thereby destroying earth ndash 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) ndash see figure 2 It is heading towards us with a velocity of 120 kmsec 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 kms 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 lsquo Big Crunchrsquo 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

2 There is a beautiful galaxy-merger movie by Dr John Dubinski at httpwwwastrosotonacukPH308galaxiesmergersMWmergempg which shows what happens when galaxies collide

5

3 Stellar Evolution

31 The Sun Our local star the sun is a typical main-sequence star of spectral type lsquoG2Vrsquo 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 ndash 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 (Ruuml) 7 1010 cm mass (Muuml) 2 1033 g mean density (ruuml) 14 gcm3

total energy output (Luuml) 3821026 Joulesec age 15 1017 sec core temperature 5 106 K surface temperature 5 103 K distance to earth 15 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 3821026 Joulesec 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

MeV726e2HeH4 01

42

11 ++++rarr νγ

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

6

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 800000 year

Assignment 4 How can we find out whether the sun has actually stopped central thermofusion in the past 800000 year

Assignment 5 Calculate the average time a photon is absorbed using that the radius of the sun = 696108 m

32 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 ndash the longer the fusion process lasts The real situation however is directly the reverse the more massive a star ndash 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 ndash B ndash A ndash F ndash G ndash K ndash M ndash R ndash N ndash 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 lsquoG2Vrsquo

Assignment 8 Using Wienrsquos 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 lsquoGrsquo for lsquogreenrsquo) 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 lsquoOh Be A Fine Girl Kiss Me Right Now ndash Ssssmackrsquo whereas some female students favor the G = lsquoGuyrsquo or lsquoGet-lostrsquo alternative

7

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 = constantT) absolute luminosity ie the cumulative energy over the entire spectrum (again using black body radiation where the luminosity L relates to the surface temperature T as Labs = constant micro T

4) 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

24 rL

relabsLπ

=

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

relLm log52minus=

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 0m14 is 119 magnitudes brighter than a Cygni (Deneb) with relative magnitude 1m33

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 ndash for instance ndash calculate the difference between ultraviolet and blue magnitude of a star U ndash B

Assignment 11 Demonstrate that the difference U ndash 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 42 light years ordm 3781013 km

8

nearby stars See figure 1 Later improvements included the difference between spectral magnitudes such as U ndash 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 ie luminosity 33 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 ie 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)

37 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

9

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 34 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 ρG where G is Newtonrsquos gravity constant G ~ 67 10-8 cm3g-1sec-2 and r the average density (see table 1) 14 gcm3 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 ndash now in ample supply because of the H-fusion ndash is fused with the remaining hydrogen to Li (lithium)

He + H Oslash Li + n

Thus a new equilibrium state has been reached which can lasts several millions years ndash 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

CHe3 126

42 rarr

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

10

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

11

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)

35 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

12

6 As this phenomenon was discovered in 1967 in the first instances the discoverers thought it was a sign of extraterrestrial intelligence

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 ndash 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 ndash 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 sect 13 Muuml 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 13 Muuml there is nothing that can halt the implosion of the star ndash 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 consideration The kinetic energy of the object at the surface of B is 21

0=E

RGmME minus= 0=

2 esckin mvE =

and when reached infinity the kinetic energy is zero According to Newtonrsquos law of gravitation the gravitational potential energy of the object at the surface of B is

and when reached infinity it is zero From the conservation of

energy find an expression the escape velocity v

kin

potEpot

esc 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 Schwarzschild-radius Calculate the Schwarzschild-for an object of one solar mass and also for an object of your own body-weight use G = 67 10-8 cm3g-1sec-2

From assignment 18 we find the so-called Schwarzschild-radius ie the radius where the escape velocity becomes the velocity of light c = 3108 ms 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 ndash therefore there is at present no theory that can adequately describe the interior of a black hole 36 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 21010 ML

14

log L Luuml

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 ndash the age the cluster formed ndash the same composition (in terms of HeH 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 (ie point A in the HRD) These results are shown in figure 7

15

Fig 7 The HRD for 10 stellar clusters At right ordinate the age in billion years of the bifurcation point from the main sequence

surface temperature (K)

L Luuml

sun

Fig 8 Abundances of chemical elements in the neighbourhood of our sun The marks are from the intensities from spectral absorption lines in the sunrsquos atmosphere the lines from meteorite and terrestrial data

16

38 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

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

Fig 9 An example of an unstable ndash but not-periodic ndash star is this massive lsquoWolf-Rayet starrsquo 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

main sequence

RR Lyrae

Cepheids

instability strip

long period variables

surface temperature

Luminosity

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 Luuml 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

0 20 40 60 80 10025

3

35

4

45

5Cepheid Luminosity-Period Law

Period [days]

log(

LLs

un)

Fig 11 Relation between luminosity and oscillation period for Cepheid type 1 variable stars

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 eg 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 ndash2 This means that electromagnetism remains 1036 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 41 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 ndash from the Greek word for lsquowanderersrsquo ndash 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 BC the Greek philosopher Aristarchos proposed a simpler ndash hence more elegant ndash 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

Copernicusrsquo 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 Occamrsquos 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 42 Newtonrsquos 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 ndash 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 Newtonrsquos law of dynamics states

)(xFp =dtd (41)

Here p = mv and F varies in space depending on the position vector x In short Newtonrsquos line of reasoning for the law of universal gravitation was

1 the force associated with gravitational action is central ie 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 lsquoA mathematical model must be as simple as possible but not too simplersquo 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 ie their masses m1 and m2 Newtonrsquos 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

rgravityr

mMG urF ˆ)(2

minus= (42)

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 ndash ie a vector of length 1 directed along vector r Note that ur can be written as ur = rr for r int 0 The proportionality between the force and the right-hand side is expressed in the constant G which in SI-units is

G = 66710-11 N m2kg2

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 = 3108 ms the proton charge e = 1610-19 C the rest mass of the electron me = 9110-31 kg and the constant of Planck h = 6610-34 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 637106 m and the acceleration of gravity at the earth surface is 98 ms2

43 Gravitational Potential Energy Since the gravitational interaction defined by equation 42 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

rgravitypot

rmMG

EurF

rˆ)(

2=minus=

part

part (43)

The solution of this equation yields

rmMGE pot minus= (44)

Here we assume the potential energy to be zero at for infinite separation

21

44 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

sumsum minus= pairsij

jiiparticles i r

mmGvmE 2frac12 (45)

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 ndash temporary ignoring all other members of the solar system Let us assume that one mass is much larger than the other M agrave m We may than approximate the energy as

rmMGmvE minus= 2frac12 (46)

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 lt 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 gt 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

45 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 43 and 44 we may write

rrMG u

rG ˆminus=

partpart

minus=φ (47)

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 eg 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 ndash given suitable equipment

Fig 3 Libration points and Roche surface in the gravitational field lines of the masses

46 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

sumsumsum minus+= pairsij

jiparticles

i

iparticles

i

irmm

GL

mp

Eη22

22 (410)

All celestial motion can now be understood as the result of dynamic motion caused by inertia rotation and the gravitational interaction ndash 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 (410) 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 agrave 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 ndash hence its name Dark matter was first identified in the halo ndash the sphere surrounding its kernel ndash of galaxies One suggestion for dark matter in these halorsquos are Massive Compact Halo Objects short MACHOrsquos Indeed MACHOrsquos 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 MACHOrsquos 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 ndash but still insufficient to account for all dark matter perhaps our understanding of physical laws is not as complete as we think 51 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 1r2-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

out

Fig 1 Orbit of a star in a galactic disk and gravitational forces from objects outside the orbit acting on the star

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 MACHOrsquos 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

52 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 kms Obviously there must be a very massive body that binds these stars in orbits Using Kepplerrsquos third law and Newtonian dynamics it is possible to estimate the mass of the central body These calculations give a mass of approximately 3106 Muuml 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 61 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 lsquouniversersquo 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 ndash a totally filled space and a partially filled universe ndash 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 ie 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 Olbersrsquos 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 ndash seemingly ndash 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 ndash or rather ndash 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 ndash in the sense of unchanging ndash 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 Olbersrsquos 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

12 After the 19th century German astronomer Heinrich Olbers 13 The last of the great Greek Astronomers who lived during the second century AD 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

27

From these arguments alone a stable static universe seems infeasible Let us now see what observations in the past century have taught us 62 Hubblersquos 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 (eg 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 ndash the closest galaxy to our own ndash 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 ie 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 420 days and an average apparent magnitude of m = +220 From this figures and the Cepheid period-luminosity relation as depicted in figure 11 from chapter 3 estimate the distance from this star ndash and so its galaxy ndash 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 ndash calculated from their redshifts using the Doppler-effect ndash 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 Hubblersquos Law

We can formulate Hubblersquos 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 Hubblersquos constant Based on currently available information the value of this constant is

H0 = 70 kmsMpc

So two galaxies separated 1 Mpc recede with a velocity of 70 kms

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 Hubblersquos law

14 The discussions whether galaxies were mere nebulae or distant and colossal star systems found its culmination in the lsquoShapley-Curtis debatersquo in the 1920s

28

Fig 1 Hubblersquos law of redshifts for 36 galaxies The redshift is calculated to the corresponding receding velocity using the Doppler effect

0 05 1 15 2 25 3 35 4 45 -50

0

50

100

150

200

250

300

350

distance in Mpc

velocity in kms

Hubbles Law of Redshifts for 36 galaxies

63 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 Hubblersquos 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 Hubblersquos 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 Hubblersquos 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 Hubblersquos law Suppose that the law of Hubble was formulated as lsquoall other galaxies are receding from us with constant velocity irrespective of the distancersquo Would such a law obey Newtonian relativity ie would it be stated equivalently disregarding your position in universe

To interpret Hubblersquos 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

29

Assignment 6 shrapnel analogy Demonstrate that in this model the individual bullets follow Hubblersquos 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

Einsteinrsquos 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 ndash 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 lsquo the most delighted moment of my lifersquo

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 ndash 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 ndash the equivalence principle ndash 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 lsquocosmological constantrsquo 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 lsquothe introduction of the cosmological constant was the biggest blunder in my lifersquo

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

lsquoThe laws of physics must be of such a nature that they apply to systems of reference in any kind of motionrsquo

30

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 ie 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 thirddimension 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 64 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 lsquoBig Bangrsquo15

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 Hubblersquos 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 Hubblersquos 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 Hubblersquos law we can estimate the time ago that a galaxy at distance d Mpc departed from our own This time is T = dv = dH0d = 1H0 Note that this time is independent from the distance d The value is the same for all galaxies Using the value of 70 kmsMpc we find

T = 14 109 years

31

15 The term was coined by the eccentric British astronomer Fred Hoyle who was skeptical towards this idea and in 1947 commented that lsquo certain American theories let us belief that the universe start in a Big Bangrsquo

Fig 4 M101 the lsquoPinwheel Galaxyrsquo 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 170000 light years It is one of the largest disk galaxies known M101 is a bright object with a magnitude of 79 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 Hubblersquos 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 lsquoherersquo lsquonowrsquo lsquopastrsquo and lsquofuturersquo 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 = 135 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 ie 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 Hubblersquos 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

32

Assignment 9 balloon analogy Demonstrate that in this model the spots follow Hubblersquos 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


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 kmsec 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 lsquoBig Crunchrsquo 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 = 02 10-27 kgm3

The Cosmic Background Radiation After about some 300000 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 Wiensrsquos 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 300000 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 2725 K after using Wienrsquos 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

33

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 300000 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 NASAs 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 10000 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 300000 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 lsquoflatrsquo 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

65 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 ie if the density is below the critical density rcrit the universe will collapse into an event that is called the lsquoBig Crunchrsquo 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

34

35

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 NASAs 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 httpwwwnasagov 2 httphubblenasagov 3 httpwwwbbccouksciencespace

4 httpwwwesaorg 5 httpwwwastrolsaumicheduCourseLabspleiadespl_introhtml

6 httpnrumianofreefrEstarssequencehtml

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 lsquoastronomyrsquo itself derives from the Greek aster meaning lsquostarrsquo and nomos meaning lsquolawrsquo 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 Einsteinrsquos 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 ndash as yet undecided ndash observatory andor 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

1






2






3






4









5

3 Stellar Evolution







MeV726e2HeH4 01

42

11 ++++rarr νγ


6














7





24 rL

relabsLπ

=




relLm log52minus=







8





9





CHe3 126

42 rarr



10






11






12








13



0=E

RGmME minus= 0=

2 esckin mvE =




kin

potEpot



14

log L Luuml

log Teff in K



15



L Luuml

sun


16




main sequence

RR Lyrae

Cepheids

instability strip


surface temperature

Luminosity


17





0 20 40 60 80 10025

3

35

4

45


Period [days]

log(

LLs

un)



18






19








)(xFp =dtd (41)




20




rgravityr

mMG urF ˆ)(2

minus= (42)


G = 66710-11 N m2kg2




rgravitypot

rmMG

EurF

rˆ)(

2=minus=

part

part (43)




21



jiiparticles i r

mmGvmE 2frac12 (45)









22


rrMG u

rG ˆminus=

partpart

minus=φ (47)





jiparticles

i

iparticles

i

irmm

GL

mp

Eη22

22 (410)



23







24



in

out




25









26







27





v = H0 d


H0 = 70 kmsMpc




28


0 05 1 15 2 25 3 35 4 45 -50

0

50

100

150

200

250

300

350

distance in Mpc

velocity in kms







29











30









T = 14 109 years

31




tPlanck = 135 10-43 s




32







rcrit = 02 10-27 kgm3



33






34

35










2






3






4









5

3 Stellar Evolution







MeV726e2HeH4 01

42

11 ++++rarr νγ


6














7





24 rL

relabsLπ

=




relLm log52minus=







8





9





CHe3 126

42 rarr



10






11






12








13



0=E

RGmME minus= 0=

2 esckin mvE =




kin

potEpot



14

log L Luuml

log Teff in K



15



L Luuml

sun


16




main sequence

RR Lyrae

Cepheids

instability strip


surface temperature

Luminosity


17





0 20 40 60 80 10025

3

35

4

45


Period [days]

log(

LLs

un)



18






19








)(xFp =dtd (41)




20




rgravityr

mMG urF ˆ)(2

minus= (42)


G = 66710-11 N m2kg2




rgravitypot

rmMG

EurF

rˆ)(

2=minus=

part

part (43)




21



jiiparticles i r

mmGvmE 2frac12 (45)









22


rrMG u

rG ˆminus=

partpart

minus=φ (47)





jiparticles

i

iparticles

i

irmm

GL

mp

Eη22

22 (410)



23







24



in

out




25









26







27





v = H0 d


H0 = 70 kmsMpc




28


0 05 1 15 2 25 3 35 4 45 -50

0

50

100

150

200

250

300

350

distance in Mpc

velocity in kms







29











30









T = 14 109 years

31




tPlanck = 135 10-43 s




32







rcrit = 02 10-27 kgm3



33






34

35










3






4









5

3 Stellar Evolution







MeV726e2HeH4 01

42

11 ++++rarr νγ


6














7





24 rL

relabsLπ

=




relLm log52minus=







8





9





CHe3 126

42 rarr



10






11






12








13



0=E

RGmME minus= 0=

2 esckin mvE =




kin

potEpot



14

log L Luuml

log Teff in K



15



L Luuml

sun


16




main sequence

RR Lyrae

Cepheids

instability strip


surface temperature

Luminosity


17





0 20 40 60 80 10025

3

35

4

45


Period [days]

log(

LLs

un)



18






19








)(xFp =dtd (41)




20




rgravityr

mMG urF ˆ)(2

minus= (42)


G = 66710-11 N m2kg2




rgravitypot

rmMG

EurF

rˆ)(

2=minus=

part

part (43)




21



jiiparticles i r

mmGvmE 2frac12 (45)









22


rrMG u

rG ˆminus=

partpart

minus=φ (47)





jiparticles

i

iparticles

i

irmm

GL

mp

Eη22

22 (410)



23







24



in

out




25









26







27





v = H0 d


H0 = 70 kmsMpc




28


0 05 1 15 2 25 3 35 4 45 -50

0

50

100

150

200

250

300

350

distance in Mpc

velocity in kms







29











30









T = 14 109 years

31




tPlanck = 135 10-43 s




32







rcrit = 02 10-27 kgm3



33






34

35










4









5

3 Stellar Evolution







MeV726e2HeH4 01

42

11 ++++rarr νγ


6














7





24 rL

relabsLπ

=




relLm log52minus=







8





9





CHe3 126

42 rarr



10






11






12








13



0=E

RGmME minus= 0=

2 esckin mvE =




kin

potEpot



14

log L Luuml

log Teff in K



15



L Luuml

sun


16




main sequence

RR Lyrae

Cepheids

instability strip


surface temperature

Luminosity


17





0 20 40 60 80 10025

3

35

4

45


Period [days]

log(

LLs

un)



18






19








)(xFp =dtd (41)




20




rgravityr

mMG urF ˆ)(2

minus= (42)


G = 66710-11 N m2kg2




rgravitypot

rmMG

EurF

rˆ)(

2=minus=

part

part (43)




21



jiiparticles i r

mmGvmE 2frac12 (45)









22


rrMG u

rG ˆminus=

partpart

minus=φ (47)





jiparticles

i

iparticles

i

irmm

GL

mp

Eη22

22 (410)



23







24



in

out




25









26







27





v = H0 d


H0 = 70 kmsMpc




28


0 05 1 15 2 25 3 35 4 45 -50

0

50

100

150

200

250

300

350

distance in Mpc

velocity in kms







29











30









T = 14 109 years

31




tPlanck = 135 10-43 s




32







rcrit = 02 10-27 kgm3



33






34

35













5

3 Stellar Evolution







MeV726e2HeH4 01

42

11 ++++rarr νγ


6














7





24 rL

relabsLπ

=




relLm log52minus=







8





9





CHe3 126

42 rarr



10






11






12








13



0=E

RGmME minus= 0=

2 esckin mvE =




kin

potEpot



14

log L Luuml

log Teff in K



15



L Luuml

sun


16




main sequence

RR Lyrae

Cepheids

instability strip


surface temperature

Luminosity


17





0 20 40 60 80 10025

3

35

4

45


Period [days]

log(

LLs

un)



18






19








)(xFp =dtd (41)




20




rgravityr

mMG urF ˆ)(2

minus= (42)


G = 66710-11 N m2kg2




rgravitypot

rmMG

EurF

rˆ)(

2=minus=

part

part (43)




21



jiiparticles i r

mmGvmE 2frac12 (45)









22


rrMG u

rG ˆminus=

partpart

minus=φ (47)





jiparticles

i

iparticles

i

irmm

GL

mp

Eη22

22 (410)



23







24



in

out




25









26







27





v = H0 d


H0 = 70 kmsMpc




28


0 05 1 15 2 25 3 35 4 45 -50

0

50

100

150

200

250

300

350

distance in Mpc

velocity in kms







29











30









T = 14 109 years

31




tPlanck = 135 10-43 s




32







rcrit = 02 10-27 kgm3



33






34

35





3 Stellar Evolution







MeV726e2HeH4 01

42

11 ++++rarr νγ


6














7





24 rL

relabsLπ

=




relLm log52minus=







8





9





CHe3 126

42 rarr



10






11






12








13



0=E

RGmME minus= 0=

2 esckin mvE =




kin

potEpot



14

log L Luuml

log Teff in K



15



L Luuml

sun


16




main sequence

RR Lyrae

Cepheids

instability strip


surface temperature

Luminosity


17





0 20 40 60 80 10025

3

35

4

45


Period [days]

log(

LLs

un)



18






19








)(xFp =dtd (41)




20




rgravityr

mMG urF ˆ)(2

minus= (42)


G = 66710-11 N m2kg2




rgravitypot

rmMG

EurF

rˆ)(

2=minus=

part

part (43)




21



jiiparticles i r

mmGvmE 2frac12 (45)









22


rrMG u

rG ˆminus=

partpart

minus=φ (47)





jiparticles

i

iparticles

i

irmm

GL

mp

Eη22

22 (410)



23







24



in

out




25









26







27





v = H0 d


H0 = 70 kmsMpc




28


0 05 1 15 2 25 3 35 4 45 -50

0

50

100

150

200

250

300

350

distance in Mpc

velocity in kms







29











30









T = 14 109 years

31




tPlanck = 135 10-43 s




32







rcrit = 02 10-27 kgm3



33






34

35


















7





24 rL

relabsLπ

=




relLm log52minus=







8





9





CHe3 126

42 rarr



10






11






12








13



0=E

RGmME minus= 0=

2 esckin mvE =




kin

potEpot



14

log L Luuml

log Teff in K



15



L Luuml

sun


16




main sequence

RR Lyrae

Cepheids

instability strip


surface temperature

Luminosity


17





0 20 40 60 80 10025

3

35

4

45


Period [days]

log(

LLs

un)



18






19








)(xFp =dtd (41)




20




rgravityr

mMG urF ˆ)(2

minus= (42)


G = 66710-11 N m2kg2




rgravitypot

rmMG

EurF

rˆ)(

2=minus=

part

part (43)




21



jiiparticles i r

mmGvmE 2frac12 (45)









22


rrMG u

rG ˆminus=

partpart

minus=φ (47)





jiparticles

i

iparticles

i

irmm

GL

mp

Eη22

22 (410)



23







24



in

out




25









26







27





v = H0 d


H0 = 70 kmsMpc




28


0 05 1 15 2 25 3 35 4 45 -50

0

50

100

150

200

250

300

350

distance in Mpc

velocity in kms







29











30









T = 14 109 years

31




tPlanck = 135 10-43 s




32







rcrit = 02 10-27 kgm3



33






34

35









24 rL

relabsLπ

=




relLm log52minus=







8





9





CHe3 126

42 rarr



10






11






12








13



0=E

RGmME minus= 0=

2 esckin mvE =




kin

potEpot



14

log L Luuml

log Teff in K



15



L Luuml

sun


16




main sequence

RR Lyrae

Cepheids

instability strip


surface temperature

Luminosity


17





0 20 40 60 80 10025

3

35

4

45


Period [days]

log(

LLs

un)



18






19








)(xFp =dtd (41)




20




rgravityr

mMG urF ˆ)(2

minus= (42)


G = 66710-11 N m2kg2




rgravitypot

rmMG

EurF

rˆ)(

2=minus=

part

part (43)




21



jiiparticles i r

mmGvmE 2frac12 (45)









22


rrMG u

rG ˆminus=

partpart

minus=φ (47)





jiparticles

i

iparticles

i

irmm

GL

mp

Eη22

22 (410)



23







24



in

out




25









26







27





v = H0 d


H0 = 70 kmsMpc




28


0 05 1 15 2 25 3 35 4 45 -50

0

50

100

150

200

250

300

350

distance in Mpc

velocity in kms







29











30









T = 14 109 years

31




tPlanck = 135 10-43 s




32







rcrit = 02 10-27 kgm3



33






34

35









9





CHe3 126

42 rarr



10






11






12








13



0=E

RGmME minus= 0=

2 esckin mvE =




kin

potEpot



14

log L Luuml

log Teff in K



15



L Luuml

sun


16




main sequence

RR Lyrae

Cepheids

instability strip


surface temperature

Luminosity


17





0 20 40 60 80 10025

3

35

4

45


Period [days]

log(

LLs

un)



18






19








)(xFp =dtd (41)




20




rgravityr

mMG urF ˆ)(2

minus= (42)


G = 66710-11 N m2kg2




rgravitypot

rmMG

EurF

rˆ)(

2=minus=

part

part (43)




21



jiiparticles i r

mmGvmE 2frac12 (45)









22


rrMG u

rG ˆminus=

partpart

minus=φ (47)





jiparticles

i

iparticles

i

irmm

GL

mp

Eη22

22 (410)



23







24



in

out




25









26







27





v = H0 d


H0 = 70 kmsMpc




28


0 05 1 15 2 25 3 35 4 45 -50

0

50

100

150

200

250

300

350

distance in Mpc

velocity in kms







29











30









T = 14 109 years

31




tPlanck = 135 10-43 s




32







rcrit = 02 10-27 kgm3



33






34

35









CHe3 126

42 rarr



10






11






12








13



0=E

RGmME minus= 0=

2 esckin mvE =




kin

potEpot



14

log L Luuml

log Teff in K



15



L Luuml

sun


16




main sequence

RR Lyrae

Cepheids

instability strip


surface temperature

Luminosity


17





0 20 40 60 80 10025

3

35

4

45


Period [days]

log(

LLs

un)



18






19








)(xFp =dtd (41)




20




rgravityr

mMG urF ˆ)(2

minus= (42)


G = 66710-11 N m2kg2




rgravitypot

rmMG

EurF

rˆ)(

2=minus=

part

part (43)




21



jiiparticles i r

mmGvmE 2frac12 (45)









22


rrMG u

rG ˆminus=

partpart

minus=φ (47)





jiparticles

i

iparticles

i

irmm

GL

mp

Eη22

22 (410)



23







24



in

out




25









26







27





v = H0 d


H0 = 70 kmsMpc




28


0 05 1 15 2 25 3 35 4 45 -50

0

50

100

150

200

250

300

350

distance in Mpc

velocity in kms







29











30









T = 14 109 years

31




tPlanck = 135 10-43 s




32







rcrit = 02 10-27 kgm3



33






34

35










11






12








13



0=E

RGmME minus= 0=

2 esckin mvE =




kin

potEpot



14

log L Luuml

log Teff in K



15



L Luuml

sun


16




main sequence

RR Lyrae

Cepheids

instability strip


surface temperature

Luminosity


17





0 20 40 60 80 10025

3

35

4

45


Period [days]

log(

LLs

un)



18






19








)(xFp =dtd (41)




20




rgravityr

mMG urF ˆ)(2

minus= (42)


G = 66710-11 N m2kg2




rgravitypot

rmMG

EurF

rˆ)(

2=minus=

part

part (43)




21



jiiparticles i r

mmGvmE 2frac12 (45)









22


rrMG u

rG ˆminus=

partpart

minus=φ (47)





jiparticles

i

iparticles

i

irmm

GL

mp

Eη22

22 (410)



23







24



in

out




25









26







27





v = H0 d


H0 = 70 kmsMpc




28


0 05 1 15 2 25 3 35 4 45 -50

0

50

100

150

200

250

300

350

distance in Mpc

velocity in kms







29











30









T = 14 109 years

31




tPlanck = 135 10-43 s




32







rcrit = 02 10-27 kgm3



33






34

35










12








13



0=E

RGmME minus= 0=

2 esckin mvE =




kin

potEpot



14

log L Luuml

log Teff in K



15



L Luuml

sun


16




main sequence

RR Lyrae

Cepheids

instability strip


surface temperature

Luminosity


17





0 20 40 60 80 10025

3

35

4

45


Period [days]

log(

LLs

un)



18






19








)(xFp =dtd (41)




20




rgravityr

mMG urF ˆ)(2

minus= (42)


G = 66710-11 N m2kg2




rgravitypot

rmMG

EurF

rˆ)(

2=minus=

part

part (43)




21



jiiparticles i r

mmGvmE 2frac12 (45)









22


rrMG u

rG ˆminus=

partpart

minus=φ (47)





jiparticles

i

iparticles

i

irmm

GL

mp

Eη22

22 (410)



23







24



in

out




25









26







27





v = H0 d


H0 = 70 kmsMpc




28


0 05 1 15 2 25 3 35 4 45 -50

0

50

100

150

200

250

300

350

distance in Mpc

velocity in kms







29











30









T = 14 109 years

31




tPlanck = 135 10-43 s




32







rcrit = 02 10-27 kgm3



33






34

35











13



0=E

RGmME minus= 0=

2 esckin mvE =




kin

potEpot



14

log L Luuml

log Teff in K



15



L Luuml

sun


16




main sequence

RR Lyrae

Cepheids

instability strip


surface temperature

Luminosity


17





0 20 40 60 80 10025

3

35

4

45


Period [days]

log(

LLs

un)



18






19








)(xFp =dtd (41)




20




rgravityr

mMG urF ˆ)(2

minus= (42)


G = 66710-11 N m2kg2




rgravitypot

rmMG

EurF

rˆ)(

2=minus=

part

part (43)




21



jiiparticles i r

mmGvmE 2frac12 (45)









22


rrMG u

rG ˆminus=

partpart

minus=φ (47)





jiparticles

i

iparticles

i

irmm

GL

mp

Eη22

22 (410)



23







24



in

out




25









26







27





v = H0 d


H0 = 70 kmsMpc




28


0 05 1 15 2 25 3 35 4 45 -50

0

50

100

150

200

250

300

350

distance in Mpc

velocity in kms







29











30









T = 14 109 years

31




tPlanck = 135 10-43 s




32







rcrit = 02 10-27 kgm3



33






34

35







0=E

RGmME minus= 0=

2 esckin mvE =




kin

potEpot



14

log L Luuml

log Teff in K



15



L Luuml

sun


16




main sequence

RR Lyrae

Cepheids

instability strip


surface temperature

Luminosity


17





0 20 40 60 80 10025

3

35

4

45


Period [days]

log(

LLs

un)



18






19








)(xFp =dtd (41)




20




rgravityr

mMG urF ˆ)(2

minus= (42)


G = 66710-11 N m2kg2




rgravitypot

rmMG

EurF

rˆ)(

2=minus=

part

part (43)




21



jiiparticles i r

mmGvmE 2frac12 (45)









22


rrMG u

rG ˆminus=

partpart

minus=φ (47)





jiparticles

i

iparticles

i

irmm

GL

mp

Eη22

22 (410)



23







24



in

out




25









26







27





v = H0 d


H0 = 70 kmsMpc




28


0 05 1 15 2 25 3 35 4 45 -50

0

50

100

150

200

250

300

350

distance in Mpc

velocity in kms







29











30









T = 14 109 years

31




tPlanck = 135 10-43 s




32







rcrit = 02 10-27 kgm3



33






34

35





log L Luuml

log Teff in K



15



L Luuml

sun


16




main sequence

RR Lyrae

Cepheids

instability strip


surface temperature

Luminosity


17





0 20 40 60 80 10025

3

35

4

45


Period [days]

log(

LLs

un)



18






19








)(xFp =dtd (41)




20




rgravityr

mMG urF ˆ)(2

minus= (42)


G = 66710-11 N m2kg2




rgravitypot

rmMG

EurF

rˆ)(

2=minus=

part

part (43)




21



jiiparticles i r

mmGvmE 2frac12 (45)









22


rrMG u

rG ˆminus=

partpart

minus=φ (47)





jiparticles

i

iparticles

i

irmm

GL

mp

Eη22

22 (410)



23







24



in

out




25









26







27





v = H0 d


H0 = 70 kmsMpc




28


0 05 1 15 2 25 3 35 4 45 -50

0

50

100

150

200

250

300

350

distance in Mpc

velocity in kms







29











30









T = 14 109 years

31




tPlanck = 135 10-43 s




32







rcrit = 02 10-27 kgm3



33






34

35







L Luuml

sun


16




main sequence

RR Lyrae

Cepheids

instability strip


surface temperature

Luminosity


17





0 20 40 60 80 10025

3

35

4

45


Period [days]

log(

LLs

un)



18






19








)(xFp =dtd (41)




20




rgravityr

mMG urF ˆ)(2

minus= (42)


G = 66710-11 N m2kg2




rgravitypot

rmMG

EurF

rˆ)(

2=minus=

part

part (43)




21



jiiparticles i r

mmGvmE 2frac12 (45)









22


rrMG u

rG ˆminus=

partpart

minus=φ (47)





jiparticles

i

iparticles

i

irmm

GL

mp

Eη22

22 (410)



23







24



in

out




25









26







27





v = H0 d


H0 = 70 kmsMpc




28


0 05 1 15 2 25 3 35 4 45 -50

0

50

100

150

200

250

300

350

distance in Mpc

velocity in kms







29











30









T = 14 109 years

31




tPlanck = 135 10-43 s




32







rcrit = 02 10-27 kgm3



33






34

35








main sequence

RR Lyrae

Cepheids

instability strip


surface temperature

Luminosity


17





0 20 40 60 80 10025

3

35

4

45


Period [days]

log(

LLs

un)



18






19








)(xFp =dtd (41)




20




rgravityr

mMG urF ˆ)(2

minus= (42)


G = 66710-11 N m2kg2




rgravitypot

rmMG

EurF

rˆ)(

2=minus=

part

part (43)




21



jiiparticles i r

mmGvmE 2frac12 (45)









22


rrMG u

rG ˆminus=

partpart

minus=φ (47)





jiparticles

i

iparticles

i

irmm

GL

mp

Eη22

22 (410)



23







24



in

out




25









26







27





v = H0 d


H0 = 70 kmsMpc




28


0 05 1 15 2 25 3 35 4 45 -50

0

50

100

150

200

250

300

350

distance in Mpc

velocity in kms







29











30









T = 14 109 years

31




tPlanck = 135 10-43 s




32







rcrit = 02 10-27 kgm3



33






34

35









0 20 40 60 80 10025

3

35

4

45


Period [days]

log(

LLs

un)



18






19








)(xFp =dtd (41)




20




rgravityr

mMG urF ˆ)(2

minus= (42)


G = 66710-11 N m2kg2




rgravitypot

rmMG

EurF

rˆ)(

2=minus=

part

part (43)




21



jiiparticles i r

mmGvmE 2frac12 (45)









22


rrMG u

rG ˆminus=

partpart

minus=φ (47)





jiparticles

i

iparticles

i

irmm

GL

mp

Eη22

22 (410)



23







24



in

out




25









26







27





v = H0 d


H0 = 70 kmsMpc




28


0 05 1 15 2 25 3 35 4 45 -50

0

50

100

150

200

250

300

350

distance in Mpc

velocity in kms







29











30









T = 14 109 years

31




tPlanck = 135 10-43 s




32







rcrit = 02 10-27 kgm3



33






34

35










19








)(xFp =dtd (41)




20




rgravityr

mMG urF ˆ)(2

minus= (42)


G = 66710-11 N m2kg2




rgravitypot

rmMG

EurF

rˆ)(

2=minus=

part

part (43)




21



jiiparticles i r

mmGvmE 2frac12 (45)









22


rrMG u

rG ˆminus=

partpart

minus=φ (47)





jiparticles

i

iparticles

i

irmm

GL

mp

Eη22

22 (410)



23







24



in

out




25









26







27





v = H0 d


H0 = 70 kmsMpc




28


0 05 1 15 2 25 3 35 4 45 -50

0

50

100

150

200

250

300

350

distance in Mpc

velocity in kms







29











30









T = 14 109 years

31




tPlanck = 135 10-43 s




32







rcrit = 02 10-27 kgm3



33






34

35












)(xFp =dtd (41)




20




rgravityr

mMG urF ˆ)(2

minus= (42)


G = 66710-11 N m2kg2




rgravitypot

rmMG

EurF

rˆ)(

2=minus=

part

part (43)




21



jiiparticles i r

mmGvmE 2frac12 (45)









22


rrMG u

rG ˆminus=

partpart

minus=φ (47)





jiparticles

i

iparticles

i

irmm

GL

mp

Eη22

22 (410)



23







24



in

out




25









26







27





v = H0 d


H0 = 70 kmsMpc




28


0 05 1 15 2 25 3 35 4 45 -50

0

50

100

150

200

250

300

350

distance in Mpc

velocity in kms







29











30









T = 14 109 years

31




tPlanck = 135 10-43 s




32







rcrit = 02 10-27 kgm3



33






34

35








rgravityr

mMG urF ˆ)(2

minus= (42)


G = 66710-11 N m2kg2




rgravitypot

rmMG

EurF

rˆ)(

2=minus=

part

part (43)




21



jiiparticles i r

mmGvmE 2frac12 (45)









22


rrMG u

rG ˆminus=

partpart

minus=φ (47)





jiparticles

i

iparticles

i

irmm

GL

mp

Eη22

22 (410)



23







24



in

out




25









26







27





v = H0 d


H0 = 70 kmsMpc




28


0 05 1 15 2 25 3 35 4 45 -50

0

50

100

150

200

250

300

350

distance in Mpc

velocity in kms







29











30









T = 14 109 years

31




tPlanck = 135 10-43 s




32







rcrit = 02 10-27 kgm3



33






34

35







jiiparticles i r

mmGvmE 2frac12 (45)









22


rrMG u

rG ˆminus=

partpart

minus=φ (47)





jiparticles

i

iparticles

i

irmm

GL

mp

Eη22

22 (410)



23







24



in

out




25









26







27





v = H0 d


H0 = 70 kmsMpc




28


0 05 1 15 2 25 3 35 4 45 -50

0

50

100

150

200

250

300

350

distance in Mpc

velocity in kms







29











30









T = 14 109 years

31




tPlanck = 135 10-43 s




32







rcrit = 02 10-27 kgm3



33






34

35






rrMG u

rG ˆminus=

partpart

minus=φ (47)





jiparticles

i

iparticles

i

irmm

GL

mp

Eη22

22 (410)



23







24



in

out




25









26







27





v = H0 d


H0 = 70 kmsMpc




28


0 05 1 15 2 25 3 35 4 45 -50

0

50

100

150

200

250

300

350

distance in Mpc

velocity in kms







29











30









T = 14 109 years

31




tPlanck = 135 10-43 s




32







rcrit = 02 10-27 kgm3



33






34

35











24



in

out




25









26







27





v = H0 d


H0 = 70 kmsMpc




28


0 05 1 15 2 25 3 35 4 45 -50

0

50

100

150

200

250

300

350

distance in Mpc

velocity in kms







29











30









T = 14 109 years

31




tPlanck = 135 10-43 s




32







rcrit = 02 10-27 kgm3



33






34

35







in

out




25









26







27





v = H0 d


H0 = 70 kmsMpc




28


0 05 1 15 2 25 3 35 4 45 -50

0

50

100

150

200

250

300

350

distance in Mpc

velocity in kms







29











30









T = 14 109 years

31




tPlanck = 135 10-43 s




32







rcrit = 02 10-27 kgm3



33






34

35













26







27





v = H0 d


H0 = 70 kmsMpc




28


0 05 1 15 2 25 3 35 4 45 -50

0

50

100

150

200

250

300

350

distance in Mpc

velocity in kms







29











30









T = 14 109 years

31




tPlanck = 135 10-43 s




32







rcrit = 02 10-27 kgm3



33






34

35











27





v = H0 d


H0 = 70 kmsMpc




28


0 05 1 15 2 25 3 35 4 45 -50

0

50

100

150

200

250

300

350

distance in Mpc

velocity in kms







29











30









T = 14 109 years

31




tPlanck = 135 10-43 s




32







rcrit = 02 10-27 kgm3



33






34

35









v = H0 d


H0 = 70 kmsMpc




28


0 05 1 15 2 25 3 35 4 45 -50

0

50

100

150

200

250

300

350

distance in Mpc

velocity in kms







29











30









T = 14 109 years

31




tPlanck = 135 10-43 s




32







rcrit = 02 10-27 kgm3



33






34

35






0 05 1 15 2 25 3 35 4 45 -50

0

50

100

150

200

250

300

350

distance in Mpc

velocity in kms







29











30









T = 14 109 years

31




tPlanck = 135 10-43 s




32







rcrit = 02 10-27 kgm3



33






34

35















30









T = 14 109 years

31




tPlanck = 135 10-43 s




32







rcrit = 02 10-27 kgm3



33






34

35













T = 14 109 years

31




tPlanck = 135 10-43 s




32







rcrit = 02 10-27 kgm3



33






34

35







tPlanck = 135 10-43 s




32







rcrit = 02 10-27 kgm3



33






34

35











rcrit = 02 10-27 kgm3



33






34

35










34

35





35





introduction to astrophysics - maastricht university · 2013-06-10 · 1. introduction astronomy is...

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