# OCR A-Level Physics/Fields, Particles and Frontiers of Physics/Structure of the Universe

## Star Formation and Life Cycle

### Formation of star

In certain areas of space, the dust and gas that is present will slowly come together, through gravitational attraction between individual atoms, to form denser clumps of matter. Given enough time, these areas will very gradually become more and more dense as more matter is attracted. This inward movement of material is called gravitational collapse.

As the gravitational force pulls more and more matter together, work is done on the particles of dust and gas, leading to an increase in kinetic energy. This results in an increase in temperature until some of the denser areas of gas become hot enough to glow. This large core of material is called a protostar

Protostars can only be detected through telescopes designed to observe infrared radiation, as the clouds of gas and dust around them (called nebulae) absorb and scatter most of the visible light.

### Nuclear Fusion

The protostar's gravitational field will continue to attract more and more matter until the temperature and pressure of the star becomes much greater. Eventually, the temperature at the core of the star will reach temperatures of up to 100 million degrees kelvin and the increased kinetic energy of the particles increases the chance that fusion reactions will begin.

There are two ways in which the fusion process can proceed. The first reaction in which 4 H nuclei may eventually result in one He nucleus is known as the proton-proton chain.

This reaction sequence is thought to be the most important one in the solar core. It is a chain reaction in the sense that two hydrogen nuclei, known as protons, start the reaction and two protons are produced at the end of the reaction, at least in one of the many variants of the proton-proton chain that exist. It is one of these variants that will be explored, it being the most common in our own star, the Sun.

In the first stage there are two distinct steps. Initially, two protons fuse to form a diproton.

H-1 + H-1 ---> He-2

The diproton then undergoes beta-decay releasing an electron neutrino and a positron as the diproton forms a deuterium nucleus. This step is extremely slow as positron emission is very rare, being controlled by the weak force. The emitted positron will usually immediately annihilate with an electron to form two gamma photons. The deuterium can fuse with another proton to form a light isotope of helium, He-3 and another gamma photon. The conversion of deuterium is very rapid as it is mediated by the strong force and estimates suggest that each deuterium nucleus exists for only four seconds before being converted.

The He-3 can exist for up to 400 years before being converted into He-4, the final form. There are other possible ways by which He-4 can be generated. The most common scenario is for two He-3 isotopes to fuse to produce one He-4 nucleus and two protons. This is known as the p-p I and is the most common with estimates suggesting that it occurs with a frequency of 83.30% in the Sun whilst other methods occur with a frequency of 16% and less.

The second reaction sequence, in which 4 H nuclei may eventually result in one He nucleus is called the CNO. The cycle generates less than 10% of the total solar energy. This involves carbon atoms which are not consumed in the overall process.

The overall effect of the fusion of hydrogen nuclei to helium nuclei is that four protons are converted into one helium-4 nucleus with the production of two gamma ray photons, two neutrinos and two positrons. During this process, enormous amounts of energy will be released. The momentum of the photons released by the fusion reactions leads to an outwards acting force called radiation pressure.

In a star of stable size, the radiation pressure and gas pressure due to the collisions of the gas atoms are in equilibrium with the gravitational force acting inwards. At this point, the star becomes a main sequence star, where it will remain for the majority of its life. Whilst a star is a main sequence star it is stable and converting hydrogen into helium through nuclear fusion.

### Development of stars

Main sequence stars make up about 90% of the stars in the universe and fuse hydrogen to form helium in their cores.

Our Sun, which is about half way through its life, is a main sequence star. The greater the mass of a star, the shorter its life tends to be and the less time it will spend on the main sequence phase due to the fact that it has a greater rate of hydrogen fusion occurring. This means it uses up its store of hydrogen quicker and dies earlier than smaller stars that burn their hydrogen more slowly and, in doing so, remain in main sequence for millions of years more.

What happens to a star at the end of its main sequence phase is dependant on its mass. The Sun will eventually follow the path of a low-mass star.

### The fate of low-mass stars

If a star is less than 1.4 times the mass of the Sun, it will move off the main sequence and first become a red giant before then turning into a white dwarf. The process begins when most of the hydrogen nuclei present in the core of a low-mass star has been fused into helium.

Life cycle of star

Nuclear fusion will stop. This means that the radiation pressure acting outwards will also stop and the star will experience a net inwards force due to gravitational attraction. The core will contract, leading to an increase in its temperature as it compresses. There is still a large quantity of hydrogen gas surrounding the core and this hydrogen will become hotter as the core continues to contract and release thermal energy. The outer layer expands to cover a greater volume than the original star, cooling and leading to the formation of a red giant.

Further contraction of the core by gravity will continue, and its temperature continues to rise. Eventually the core becomes hot enough for fusion of helium nuclei to take place, leading to the production of heavier elements, including carbon and oxygen. Enormous amounts of energy are released when the fusion reactions take place, increasing the radiation pressure outwards.

Once the fusion of helium finishes, a low-mass star is not hot enough for further fusion reactions to occur so fusion stops. The star becomes unstable and begins to collapse again. At this stage, the outer layers of gas may be ejected into space forming a planetary nebula.

The rest of the star continues to collapse under its own mass and to heat up until it reaches a point where it can collapse no further. What is left is a very hot, dense core called a white dwarf. No further fusion reactions take place but the star continues to radiate energy as the photons produced from past fusion reactions leak away. Eventually, this white dwarf will gradually cool down to a surface temperature of just a few kelvin.

### Electron degeneracy pressure

When matter is compressed into a very small volume such as the very dense core of a collapsing star, the electrons are no longer free to move about between energy levels. As the star contracts, the compression forces electrons in neighbouring atoms into the lowest energy levels first and then into higher and higher energy levels once the lowest unoccupied energy levels are filled.

It's not possible for two or more electrons to occupy identical states in an energy level at the same time so when all of the available electron states are full, it is not possible to add another electron to a given volume. It is as if the electrons exert a repulsive outwards force. This is known as electron degeneracy pressure.

If a star is not too massive, the electron degeneracy pressure prevents further gravitational collapse and a stable white dwarf is formed. The electron degeneracy pressure is only sufficient to prevent collapse of the star if the stellar remnant is less than a maximum mass called the Chandrasekhar limit

## High-mass stars

When larger, more massive stars move off the main sequence they are already much brighter than lower-mass stars. The core contracts under gravity and heats up while the outer layers expand to many times the star's original size to become a red super giant. As the core collapses and heats up, further nuclear fusion reactions can occur, with fusion of heavier and heavier elements possible at higher temperatures and pressures. In each stable fusion phase, the degeneracy pressure of electrons and radiation pressure prevent gravitational collapse.

Fusion continues until an iron core builds up; this core then collapses. The fate of the red super giant then depends on the mass of the remaining core, If the mass of its core is less than 1.4 solar masses (the Chandrasekhar limit) the star will remain as a white dwarf. If the mass of the star's core is greater than this, the electron degeneracy pressure will not be sufficient to support the core and the core will undergo gravitational collapse.

### Supernovae

The final collapse of the iron core may only take a few days, during which the loss of gravitational potential energy produces intense heating. During the final seconds of the collapse of a red super giant, the immense gravitational pressure forces protons and electrons in the iron to combine to become neutrons. This triggers an explosive blowing out of the outer shell.

The huge release of energy is called a supernova. Although elements heavier than iron cannot be produced by fusion in stars in a supernova elements heavier than iron can be formed when the remaining heavy nuclei capture, or fuse with, a neutron. The Earth and all the elements in it are the remains of a supernova explosion.

### Neutron stars

Under certain conditions the extremely dense collapsed neutron core can remain intact after a supernova explosion. This is a neutron star. Its density will be such that a neutron star of mass equal to that of the Sun would have a diameter of only 30km. Some neutron stars rotate rapidly and emit highly directional bursts of EM waves. The magnetic field of the neutron star must be large, and the frequency at which the pulses of radiation are emitted is assumed to be the rate at which the star, and its associated magnetic field rotates.

The star sends out this radiation somewhat like the revolving beam of a lighthouse, Such neutron stars are called pulsars.

### Black holes

A further stage of development is possible when the core mass is greater than three of four solar masses. Theoretically, the pressure on the core could become so large that the neutron star would collapse to a point at which the density would become infinite. Whether normal laws of physics apply at this stage is debatable.

### Development of stars

Main sequence stars make up about 90% of the stars in the universe and fuse hydrogen to form helium in their cores. Our Sun, which is about half way through its life, is a main sequence star. The greater the mass of a star, the shorter its life tends to be and the less time it will spend on the main sequence.

What happens to a star at the end of it hydrogen fusion phase is dependant on its mass. The Sun will follow the eventual path of a low-mass star.

### Hertzsprung-Russell diagram

The Hertzsprung-Russell diagram shows the position of stars on a scatter graph based on their luminosities or magnitude and their temperatures. It was devised by the Danish astronomer Ejnar Hertzsprung and the American astronomer Henry Norris Russell between 1911 and 1913. You need to learn the general positions of the graph.

Magnitude or brightness is an important factor in the classification of stars and can be measured in two ways. One is by judging the brightness of a star simply by looking at it. This has many flaws given the fact that stars farther away will appear dimmer though they may be brighter than stars that are closer. The sun ,for example, is the brightest object in our sky but is by no means the brightest object in our galaxy. This measure is known as apparent magnitude. Astronomers can also determine the rate of power output, which is linked to luminosity, of stars at a constant distance from the star of 10 parsecs. This provides a more accurate measure of the true brightness of a star and this is the data used in the H-R diagram. This measure is known as the absolute magnitude.

Astronomers can also classify stars according to their temperature. Stars of different surface temperatures emit the majority of their radiation across a certain range of wavelengths with the most common wavelength known as the peak wavelength. If the peak wavelength of a star is identified, its maximum surface temperature can be as well.

Both these data can be plotted on a graph with the y-axis as absolute magnitude and the x-axis as temperature. There is one oddity; the temperature scale is inverted so that the temperature decreases as it proceeds from the origin.

The diagram, when the axes are arranged as such, shows a trend with main sequence stars forming a line diagonally across the diagram and two clusters above and below the line, one consisting of white dwarves below the line and the other consisting of giants; red, red super and other giant stars. We can predict the temperature of star knowing only its magnitude and vice-versa using the H-R diagram.

### Spectrum of the sun

It was thought that visible light would be a continuous spectrum however a physicist discovered dark lines on the Sun's spectrum where certain frequencies are missing.

### Energy levels in atoms and the production of spectra

The Danish physicist Niels Bohr explained the existence of spectral lines through the use of the new photon model of electromagnetic radiation proposed by Max Planck and Albert Einstein. Bohr proposed electrons in atom could only behave in certain ways:

• Electrons orbit the nucleus
• The electrons can only occupy certain orbits, called energy levels
• Electrons can only gain and lose energy by moving from one allowed energy level to another, absorbing or emitting EM radiation with a frequency, f, determined by the energy difference of the levels according to the Planck relation for the photon's energy:

${\displaystyle hf=\Delta \ E}$  or ${\displaystyle hf=E_{1}-E_{2}}$

Where ${\displaystyle E_{1}}$  is the energy associated with the energy level the electron has left, and ${\displaystyle E_{2}}$  is the energy associated with the level the electron moves to. The energy difference between two energy levels can also be written as:

${\displaystyle \Delta \ E={{hc} \over \ \lambda \ }}$

### Negative energy values for energy levels

An electron's energy is taken to be zero when the electron is a very long way from the atom's nucleus. This is similar to how we define gravitational potential to be zero at infinity. As an electron moves towards the nucleus form very far away, its energy decreases below zero so energy levels inside an atom have negative values.

When the electron us at its lowest energy level, we say that this is the ground state. Removing an electron from an atom completely is called ionisation.

### Emission spectra

An emission line spectrum is produced when an excited electron in an atom, favouring greater stability, moves from a higher to a lower energy level and emits a photon with an energy corresponding to the difference between these energy levels. The spectrum is a dark background with coloured lines showing the frequencies of light emitted by the electrons. Hot gases produce emission line spectra. Each of the lines on the emission spectra for hydrogen and helium shown, corresponding to a different energy level transition within the atom. Each element's emission line spectrum is unique and acts as a fingerprint for the element.

### Emission of light from stars

Fusion reactions in a star's core produce photons of electromagnetic radiation which move upwards through the layers of gas surrounding the core. The photons are constantly absorbed by atoms in these gases, which then emit photons of many different frequencies, in random directions. The electromagnetic radiation coming from a star's inner region in therefore a continuous spectrum of wavelengths from radio through visible light to gamma rays.

### Absorption spectra from the Sun and other stars

The dark lines in the Sun's spectrum are caused by the presence of certain elements including hydrogen and helium in the Sun's gaseous atmosphere. The Sun's atmosphere above its visible surface is relatively cool compared with its inner layers. The Sun's atmosphere is not hot enough to produce emission lines.

When an atom in a gas absorbs a photon of the correct energy, an electron moves from a lower energy level to a higher one. Since the energy levels are discrete, only certain frequencies will be absorbed. The particular wavelengths of light corresponding to these transitions are then missing from the continuous spectrum, this is an absorption line spectrum.

### Transmission diffraction grating

Dispersion, the separation of visible light into a spectrum, may be accomplished by means of a prism or a transmission diffraction grating. Diffraction gratings are essential to analyse the wavelengths of light from a star because the angular dispersion or separation of colours is much greater than with a prism, allowing for more accurate measurements.

Light transmitted through multiple slits of the grating will be diffracted at different angles based on the wavelength of the incident light and the separation of the slits. For each wavelength of light from the source, the diffracted beams interfere. When the path difference between light from adjacent slits is a whole number of wavelengths, then the waves are in phase and constructive interference occurs. When two waves have a phase difference of 180 degrees. destructive interference occurs and the resultant wave has zero amplitude.

There are a number of angles at which light is seen, these are called interference lines or maxima. Between the maxima there is darkness. If the incident light is monochromatic, each maximum is a single line of that colour. If the incident light has a spectrum of wavelengths, each wavelength interferes separately so each maximum is a spectrum.

The wavelength of the light source can be determined from the equation ${\displaystyle n\lambda =dsin\theta }$

where ${\displaystyle n}$  is the order of the maximum, ${\displaystyle \lambda }$  is the wavelength of the incident monochromatic light, ${\displaystyle d}$  is the separation of the slits in the grating and ${\displaystyle \theta }$  is the angle that the beam makes with the grating. When the path difference is zero, there is a central line with no spectrum.

## Wein's law and Stefan's law

### The colour-temperature relationship

The peak wavelength of electromagnetic radiation emitted by a star depends on its temperature.

The spectrum of the Sun shows that it emits most of its radiation in a continuous wavelength band from 100 to 4000 nanometres. Other stars shine brighter in infrared or ultraviolet wavelengths because they have different temperatures.

### Wien's displacement law

${\displaystyle \lambda _{max}}$  is the wavelength that corresponds to the peak intensity or power of the star's emission.

${\displaystyle \lambda _{max}\propto {1 \over T}}$  or ${\displaystyle \lambda _{max}T=}$  constant = 2.89 x ${\displaystyle 10^{-3}}$  mK

This is used to estimate the maximum surface temperature of a star from the star's spectrum. It is important to note here that the letters mK are units meter kelvin, not milli-kelvin.

Wien's law is an empirical law, meaning it was based on available data.

### Stefan's law

The luminosity of a star is the total energy that the star emits per second, or the power. A star's luminosity, ${\displaystyle L}$ , is related to its surface area and temperature by the equation:

${\displaystyle L=4\pi {r^{2}}\sigma {T^{4}}}$

Where ${\displaystyle r}$ is the star's radius and the symbol ${\displaystyle \sigma }$ is the Stefan constant and has a value of 5.67 x ${\displaystyle 10^{-8}}$  W ${\displaystyle m^{-2}K^{-4}}$ .

Important relationships to note is that luminosity is proportional to the surface area of a star so the larger the surface area, the more energy it will emit per unit time. Luminosity is also proportional to the fourth power of temperature so the temperature increases rapidly as the power output of a star increases.

If two stars have the same temperatures, the star that appears brighter has the largest diameter. This is important when considering how the size of a star must change as it goes through its life cycle.

### Problem when using these laws

We have seen that ${\displaystyle \lambda _{max}\propto {1 \over T}}$  and ${\displaystyle L=4\pi {r^{2}}\sigma {T^{4}}}$  allow a star's temperature and size to be estimated from its spectrum and luminosity. However there are some difficulties in obtaining highly accurate results due to:

• The Earth's atmosphere only allows certain wavelengths of electromagnetic radiation to pass through.
• Dust and human light pollution also affect the quality of the data we collect.
• Detectors are often wavelength sensitive. Some detectors do not respond to some wavelengths, which could provide inaccurate values for ${\displaystyle \lambda _{max}}$ of a star which radiates mainly in the wavelength region.

## Astronomical distances

The astronomical unit of distance is the mean distance from the centre of the Earth to the centre of the Sun. The astronomical unit is used for studies of the solar system. Its value has been measured as 1 astronomical unit (AU) = 1.496 x ${\displaystyle 10^{11}}$ m.

### The parsec (pc)

The distance to a star can be calculated using the stellar parallax method by observing how its position against a fixed background of stars changes over time. The principal of parallax is based on the fact that the apparent position of a star appears to change when viewed from different positions. The position of a star is recorded every six months, when the Earth is on the opposite side of its orbit and the angle by which the star appeard to have shifted is measured. This is called a parallax angle and is calculated in arc angles which are merely sub divisions of a degree. One arcminute is a sixtieth of a degree and one arcsecond is a sixtieth of an arcminute, hence an arcsecond is one three thousand six hundredths of a degree.

The unit of the parsec becomes untenable when trying to gauge the distance of any stars outside our own galaxy simply because they are too far away. The parallax angle becomes too small a fraction of a degree to accurately measure.

The term parsec comes from an abbreviation of parallax and arc second.

A star is exactly 1 parsec away from Earth when the angle of parallax subtended by the radius of the Earth's orbit is 1 arc second or ${\displaystyle 1 \over 3600}$ ths of a degree.

The parsec is much greater in size than the astronomical unit, so we use it to state the distance between the Earth and stars, or neighbouring galaxies. At the scale of the solar system, the parsec would be inappropriate as it would be too large, over 20,000 times greater than the size of the AU.

For a right angled triangle, ${\displaystyle tan\ p={r \over d}}$ , for small angle approximations we use ${\displaystyle tan\ p\approx p}$ . It can be shown that the relationship between the distance to a star from Earth and the angle of stellar parallax is given by ${\displaystyle p={1 \over d}}$ , where ${\displaystyle p}$  is the stellar parallax measured in seconds of arc and ${\displaystyle d}$  is the distance in parsecs.

### The light-year (ly)

The light-year is the distance that light will travel in one year in a vacuum. It is a measurement of distance, not time.

Light travels at a velocity of 3 x ${\displaystyle 10^{8}}$  meters per second. In one year it will travel 9.46 x ${\displaystyle 10^{13}}$ metres.

## The Doppler effect and red shift

### The Doppler effect

When a siren from a moving ambulance or fire engine is near us, the frequency of the sound that the observer detects will change depending on whether the vehicle is moving towards us to away from us. The Doppler effect will not only work for sound waves, but also electromagnetic waves like, for example, a galaxy moving towards or away from the Earth.

### Doppler shift for electromagnetic radiation

For a light source that is moving towards us, let us define velocity as ${\displaystyle v}$ , the speed of light as ${\displaystyle c}$ , the source's change in wavelength as ${\displaystyle \Delta \lambda }$  and its change in frequency as ${\displaystyle \Delta f}$ . The wavelength of the stationary source we shall call ${\displaystyle \lambda }$  and the frequency of the stationary source will be called ${\displaystyle f}$ .

For an electromagnetic wave produced by a stationary source the wavelength is given by ${\displaystyle \lambda ={c \over f}}$ . However, if the source is moving towards us at speed ${\displaystyle v}$ , then it will have moved an additional distance towards us of ${\displaystyle v \over f}$ . This means that the difference between the successive crests will be ${\displaystyle {c \over f}}$  - ${\displaystyle v \over f}$ .

If we consider the changes in wavelength and frequency for a source of electromagnetic radiation moving relative to an observer, then provided that ${\displaystyle v}$  is much less than the speed of light, ${\displaystyle c}$ , then:

${\displaystyle {\Delta \lambda \over \lambda }\approx {v \over c}}$  and ${\displaystyle {\Delta f \over f}\approx {v \over c}}$ .

### Spectral lines and red shift

When spectral lines are examined for stars that are travelling away from us, the pattern of lines is shifted towards the red end of the spectrum. This is known as red shift. The waves of receding objects are stretched as they travel to Earth which causes an increase in the wavelength of the radiation which pushes it to the redder end of the spectrum. Light with larger wavelength will shift more than light with smaller wavelengths.

The same phenomenon occurs for objects approaching Earth. The wavelength of the radiation emitted is squashed, thereby decreasing its wavelength and shifting it towards the blue end of the spectrum. This is known as blue shift.

Redshift is an important proof of the Big Bang theory in that the fact that the majority of spectra from galaxies show redshift must mean they are all moving away from us. This indicates that they must have been closer together at some point and are being pulled apart from each other by expanding space-time. The idea of space expanding is a central basis of the Big Bang theory and thus the one confirms, or at least supports, the other.

### Hubble's law

Edwin Hubble showed that the universe is expanding, he noticed that, for the majority of galaxies observed, the spectral lines were shifted towards the red end of the spectrum, meaning most of the galaxies were moving away from us.

Hubble's law states that the recessional velocity of a galaxy is proportional to its distance from the Earth so that galaxies farther away are receding quicker than those nearest us. This can be summarised in the equation, ${\displaystyle v\approx H_{o}d}$ . Where ${\displaystyle v}$  is the recessional velocity of a galaxy in ${\displaystyle km\ s^{-1}}$ , ${\displaystyle d}$  is the distance of the galaxy from the Earth in megaparsecs, and ${\displaystyle H_{0}}$  is the Hubble constant, being approximately 75 ${\displaystyle km\ s^{-1}}$ ${\displaystyle Mp{c^{-1}}}$ .

The S.I. units of the Hubble constant is per second. Hence, the inverse of Hubble's constant has the units seconds. The value of the inverse of Hubble's constant is around 4.55x1017 seconds which, in years, is around 14 billion. This bears remarkable similarity to the estimated age of the Universe which is around 13.8 billion years. The inverse of the Hubble constant, also known as Hubble time, is the age of the Universe given that the rate of expansion is linear, which it is not.

## Microwave background radiation and the cosmological principle

The serendipitous discovery of CMBR by American radio astronomers Arno Penzias and Robert Wilson was the culmination of work initiated in the 1940s, and earned the discoverers the 1978 Nobel Prize in Physics. They had discovered cosmic microwave background radiation - the echo of the Big Bang.

The discovery of CMBR is landmark evidence of the Big Bang origin of the universe. When the universe was young, before the formation of stars and planets, it was denser, much hotter, and filled with a uniform glow from a white-hot fog of hydrogen plasma. As the universe expanded, both the plasma and the radiation filling it grew cooler. When the universe cooled enough, protons and electrons combined to form neutral hydrogen atoms. Unlike the uncombined protons and electrons, these newly conceived atoms could not absorb the thermal radiation, and so the universe became transparent instead of being an opaque fog. Cosmologists refer to the time period when neutral atoms first formed as the recombination epoch, and the event shortly afterwards when photons started to travel freely through space rather than constantly being scattered by electrons and protons in plasma is referred to as photon decoupling. The photons that existed at the time of photon decoupling have been propagating ever since, though growing fainter and less energetic, since the expansion of space causes their wavelength to increase over time (and wavelength is inversely proportional to energy according to Planck's relation), so that we are today left with the low-energy remnants that now occupy the microwave region of the EM spectrum.

The temperature of this radiation is around 2.7 K which is likely to decrease as the Universe expands and the radiation cools. It is also very uniformly distributed across the Universe with only tiny fluctuations in its intensity but even this corresponds to models of hot gas expanding to sizes of the observable Universe and these fluctuations are thought to be necessary for the formation of galaxies.

The cosmological principle states that on a large scale, the universe is uniform, this means that the universe is:

• Isotropic - the same in all directions. Although there are galaxies and clusters of galaxies that clump together, on the larger scale the distribution of matter seems uniform.
• Homogenous - of uniform density.
• subject everywhere to the same physical laws and models that apply here on Earth.